#6612567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	#24397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fh,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,e^d=ehi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257562, #72257577, #72264263
maximal normal subgroup of	#72257562, #72257577, #72264263
maximal quotient of	#72264768

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519763);