#6745575

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35513	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,31)(28,30)(29,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(19,27)(24,31)(28,29)(30,32);(11,15)(13,16)(18,23)(20,21)(24,30)(25,26)(28,29)(31,32);(2,6)(5,7)(17,19,22,27)(18,24,26,31)(20,29,21,28)(23,30,25,32);(3,8)(5,7)(11,15)(13,16)(17,20,22,21)(18,25,26,23)(19,28,27,29)(24,30,31,32)

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257512, #72257577, #72264348
maximal normal subgroup of	#72257512, #72257577, #72264348
maximal quotient of	#72307875

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6652771);