#6810141

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

2220 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=i,e^2=fh,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, 6717337);