#6576262

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483458);