#7021619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

2235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,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, 6928815);