#6460249

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367445);