#152491

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#120
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12976
prime radical	#525	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

317 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 108 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 — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217181, #72224993, #72225001, …
maximal normal subgroup of	#72217181, #72224993, #72225001, …
maximal quotient of	C₂×#152491

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59687);