#6811326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2039 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=hi,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718522);