#512548

order	512
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂²×C₄²
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂²×#346
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups; only the 432 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 432 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,g^2=h,h^2=i,b^a=bh,f^a=fi,e^b=ei,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217181, #72217183, #72217199, …
maximal normal subgroup of	#72217181, #72217183, #72217199, …
maximal quotient of	#72231854, #72231855, #72231857, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419744);