#6682219

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2291 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:

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

Polycyclic presentation:

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

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, 6589415);