#6815003

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1981 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)(6,9)(7,11)(13,14)(15,20)(16,26)(17,27)(18,23)(19,21)(22,24)(25,28);(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,24,25,17)(16,21,19,26)(20,27,28,22);(1,2,3,4)(13,15,19,22,18,25,16,27)(14,20,21,24,23,28,26,17);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,21,23,26)(15,22,25,27)(17,28,24,20);(6,9)(10,12)(13,17,19,20,18,24,16,28)(14,22,21,25,23,27,26,15)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722199);