#6816410

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1842 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)(7,11)(8,10)(9,12)(13,14,20,24)(15,21,26,28)(16,18,23,19)(17,22,25,27);(2,3)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,27)(17,18)(19,25)(20,26)(22,23)(24,28);(1,2,4,3)(6,10)(7,12)(13,16,19,14,20,23,18,24)(15,22,25,28,26,27,17,21);(6,10)(9,11)(13,17,20,25)(14,22,24,27)(15,18,26,19)(16,21,23,28);(1,3,4,2)(13,18,20,19)(14,23,24,16)(15,25,26,17)(21,27,28,22)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723606);