#6682331

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

2273 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589527);