#6682455

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6589651);