#6460427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
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	#34743	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1996 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)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,22,27,24)(20,28,30,31)(23,29,32,26);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,27)(18,22,25,24)(20,26,30,29)(23,31,32,28);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,19,26,21,30,27,29)(18,23,24,28,25,32,22,31);(1,4,5,3)(2,8,6,7)(11,15)(13,16);(17,19,21,27)(18,22,25,24)(20,29,30,26)(23,28,32,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgh,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^c=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, 6367623);