#6816855

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2150 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)(5,6,9,11)(7,12,8,10)(13,14,17,19)(15,20,24,32)(16,26,29,21)(18,22,31,33)(23,36,39,34)(25,38,28,35)(27,37,30,41)(40,43,42,44);(5,6)(7,10)(8,12)(9,11)(14,19)(15,23)(16,27)(18,31)(20,34)(21,37)(24,39)(26,41)(29,30)(32,36)(35,38)(40,42);(1,2,3,4)(13,15,18,23,17,24,31,39)(14,20,22,36,19,32,33,34)(16,28,30,42,29,25,27,40)(21,38,37,43,26,35,41,44);(5,7,9,8)(6,10,11,12)(14,19)(15,24)(21,26)(22,33)(23,39)(25,28)(37,41)(40,42);(5,8,9,7)(6,10,11,12)(13,16,18,30,17,29,31,27)(14,21,22,37,19,26,33,41)(15,25,23,40,24,28,39,42)(20,35,36,44,32,38,34,43)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6724051);