#6468991

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437412
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24214	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,29)(21,28)(22,27)(24,32)(25,31)(26,30);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(19,27)(23,31)(26,32)(28,29);(10,14)(13,16)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,26,31,32);(1,4,5,3)(2,6,7,8)(18,25)(23,31)(24,30)(26,32)

Polycyclic presentation:

2^9|c^2=g,d^2=i,e^2=ghi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308792, #72308988, #72309037
maximal normal subgroup of	#72308792, #72308988, #72309037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376187);