#6816922

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2116 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei

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, 6724118);