#137159

order	512
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1595
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9017
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

685 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=ehi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215183, #72217918
maximal normal subgroup of	#72217918
maximal quotient of	#72215049, #72215055, #72215063

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44355);