#6557416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257627, #72264068, #72264310
maximal normal subgroup of	#72257627, #72264068, #72264310
maximal quotient of	#72265629

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6464612);