#6806845

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2074 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)(7,11)(8,10)(9,12)(13,14)(15,25)(16,18)(17,27)(19,24)(20,23)(21,26)(22,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,27)(17,22)(18,20)(19,26)(23,28)(24,25);(2,4)(13,16,17,21,19,28,20,25)(14,22,27,15,24,18,23,26);(6,10)(9,11)(13,17,19,20)(14,23,24,27)(15,22,26,18)(16,21,28,25);(7,12)(9,11)(13,18,20,15,19,22,17,26)(14,16,23,25,24,28,27,21)

Polycyclic presentation:

2^9|c^2=hi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6714041);