#6816139

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

2287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,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,e^c=ei

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, 6723335);