#483327

order	512
nilpotency class	4
isoclinism	isoclinic class of #33404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#106
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂²×#334
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#131 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups; only the 352 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 352 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,13,8,14,30,41,9,15,29,12,7,32,5,28)(3,16,19,34,21,43,18,35,22,40,31,56,20,50,49,10)(4,11,25,38,27,39,51,60,23,36,52,37,26,55,24,53)(17,42,46,58,48,61,45,57,44,59,54,64,47,63,62,33);(1,3,5,18,7,20,29,19,9,22,30,49,8,21,6,31)(2,10,12,16,14,34,28,43,15,35,13,40,32,56,41,50)(4,17,24,45,26,47,52,46,23,44,51,62,27,48,25,54)(11,33,37,42,39,58,53,61,36,57,38,59,55,64,60,63);(1,4)(2,11)(3,17)(5,24)(6,25)(7,26)(8,27)(9,23)(10,33)(12,37)(13,38)(14,39)(15,36)(16,42)(18,45)(19,46)(20,47)(21,48)(22,44)(28,53)(29,52)(30,51)(31,54)(32,55)(34,58)(35,57)(40,59)(41,60)(43,61)(49,62)(50,63)(56,64);(4,23)(11,36)(17,44)(24,51)(25,52)(26,27)(33,57)(37,38)(39,55)(42,59)(45,62)(46,54)(47,48)(53,60)(58,64)(61,63)

Polycyclic presentation:

2^9|a^2=f,b^2=e,e^2=g,f^2=h,g^2=i,h^2=i,b^a=be,e^a=eg,g^a=gi,f^b=fi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217272, #72217274, #72224948, …
maximal normal subgroup of	#72217272, #72217274, #72224948, …
maximal quotient of	#72231928, #72231931, #72231934, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390523);