#6444892

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2183 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)(7,10)(8,9)(11,12)(13,14,18,20)(15,24,26,21)(16,22,25,28)(17,27,19,23);(6,9)(10,12)(14,20)(15,25)(16,26)(17,19)(21,28)(22,24);(1,2,3,4)(13,15)(14,21)(16,19)(17,25)(18,26)(20,24)(22,27)(23,28);(13,16,18,25)(14,22,20,28)(15,19,26,17)(21,27,24,23);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,20,27)(15,25,26,16)(21,28,24,22)

Polycyclic presentation:

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

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, 6352088);