#6627432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=h,d^2=h,e^2=h,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=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6534628);