#6816199

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1850 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)(7,11)(8,10)(9,12)(13,14)(15,24)(16,18)(17,27)(19,23)(20,22)(21,25)(26,28);(5,7,8,12)(6,9,10,11)(13,15,19,25)(14,21,23,24)(16,20,26,17)(18,27,28,22);(1,2,3,4)(6,10)(7,12)(13,16,20,25,19,26,17,15)(14,18,22,21,23,28,27,24);(6,10)(9,11)(13,17,19,20)(14,22,23,27)(15,16,25,26)(18,24,28,21);(13,18,20,21,19,28,17,24)(14,16,22,25,23,26,27,15)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723395);