#6491283

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3143 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398479);