#6682372

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,26)(17,22)(18,32)(19,23)(21,28)(24,35)(25,33)(27,39)(29,40)(30,36)(31,37)(34,41)(38,44)(42,43);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,23,32)(16,24,29,28)(17,25,31,30)(21,26,35,40)(22,33,37,36)(27,38,42,41)(34,39,44,43);(2,4)(13,16,18,28,19,29,15,24)(14,21,20,26,23,35,32,40)(17,27,30,41,31,42,25,38)(22,34,33,39,37,44,36,43);(6,10)(9,11)(13,17)(14,22)(15,25)(16,27)(18,30)(19,31)(20,33)(21,34)(23,37)(24,38)(26,39)(28,41)(29,42)(32,36)(35,44)(40,43);(2,4)(7,12)(9,11)(13,15,19,18)(14,20,23,32)(16,24,29,28)(17,30,31,25)(21,26,35,40)(22,36,37,33)(27,41,42,38)(34,43,44,39)

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6589568);