#131321

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1613
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8403
prime radical	#60749	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

737 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217918
maximal normal subgroup of	#72217918
maximal quotient of	#72216369, #72216380, #72216384

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38517);