#153552

order	512
nilpotency class	6
isoclinism	isoclinic class of #153549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13584
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	#492	C₂³
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3152 ◁ G
LECS	G ▷ #492 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

259 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,14,17,38)(4,9)(5,22,25,10)(6,12,28,11)(7,13)(8,16,30,49)(15,42,46,63)(18,44,27,43)(19,51,53,34)(20,36,55,35)(21,37)(23,32)(24,56,58,39)(26,40,59,41)(29,48,33,45)(31,50)(47,60,54,64)(52,61)(57,62);(1,3,7,18)(2,8,13,30)(4,15,21,47)(5,23,26,50)(6,27,28,17)(9,33,37,29)(10,38,41,43)(11,32,12,31)(14,22,44,40)(16,25,49,59)(19,52,24,57)(20,54,55,46)(34,63,39,64)(35,61,36,62)(42,51,60,56)(45,53,48,58);(1,4,7,21)(2,9,13,37)(3,15,18,47)(5,24,26,19)(6,20,28,55)(8,29,30,33)(10,39,41,34)(11,35,12,36)(14,42,44,60)(16,48,49,45)(17,46,27,54)(22,56,40,51)(23,57,50,52)(25,58,59,53)(31,61,32,62)(38,63,43,64)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=ghi,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=chi,d^b=dgh,e^b=efgi,f^b=fhi,g^b=gi,d^c=di,e^d=egh,f^e=fhi,g^e=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306810, #72306811, #72306816
maximal normal subgroup of	#72306810, #72306811, #72306816

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60748);