#6671063

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1893 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)(5,6,8,9)(7,11,12,10)(13,14,18,19)(15,20,24,28)(16,25,27,21)(17,22,23,26);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,25)(18,24)(19,28)(21,23)(22,27);(2,4)(6,9)(10,11)(13,16)(14,21)(15,23)(17,24)(18,27)(19,25)(20,22)(26,28);(5,7,8,12)(6,10,9,11)(13,14,18,19)(15,20,24,28)(16,21,27,25)(17,26,23,22);(13,17,18,23)(14,22,19,26)(15,16,24,27)(20,25,28,21)

Polycyclic presentation:

2^9|a^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,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, 6578259);