#6440102

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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

2211 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)(7,10)(8,9)(11,12)(13,14,18,17)(15,22,19,20)(16,24,25,28)(21,27,26,23);(1,2)(3,4)(6,9)(10,12)(13,15,18,19)(14,20,17,22)(16,23,25,27)(21,24,26,28);(2,4)(13,16,18,25)(14,21,17,26)(15,23,19,27)(20,24,22,28);(2,4)(13,17,18,14)(15,22,19,20)(16,21,25,26)(23,24,27,28);(5,7,8,11)(6,10,9,12)(13,15,18,19)(14,22,17,20)(16,23,25,27)(21,28,26,24)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,g^c=gi,e^d=ei

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, 6347298);