#6682769

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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

2344 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)(14,19)(15,20)(21,26)(23,27);(5,7)(6,9)(8,12)(10,11)(13,14)(15,22)(16,23)(17,19)(18,26)(20,28)(21,25)(24,27);(1,2,3,4)(13,15)(14,16)(17,20)(18,27)(19,24)(21,28)(22,26)(23,25);(6,10)(9,11)(14,19)(16,24)(21,26)(22,28);(7,12)(9,11)(13,16,17,24)(14,20,19,15)(18,28,25,22)(21,23,26,27)

Polycyclic presentation:

2^9|c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6589965);