#6811555

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2060 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6718751);