#6814467

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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

1937 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)(7,11)(8,10)(9,12)(15,24)(17,28)(19,26)(21,25);(5,7)(6,9)(8,12)(10,11)(13,14)(15,17)(16,20)(18,22)(19,25)(21,26)(23,27)(24,28);(1,2,3,4)(13,15,16,25,18,24,27,21)(14,19,23,17,22,26,20,28);(6,10)(9,11)(13,16,18,27)(14,20,22,23)(15,25,24,21)(17,19,28,26);(7,12)(9,11)(13,17,16,26,18,28,27,19)(14,21,23,15,22,25,20,24)

Polycyclic presentation:

2^9|c^2=ghi,d^2=i,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,e^d=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, 6721663);