#6811428

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2088 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)(13,14,17,21)(15,23,25,19)(16,27,30,20)(18,22,31,36)(24,37,38,32)(26,33,29,35)(28,41,40,34)(39,43,42,44);(5,7,10,6)(8,12,9,11)(15,24)(16,28)(18,31)(19,32)(20,34)(22,36)(23,37)(25,38)(26,29)(27,41)(30,40)(33,35);(2,4)(13,15,18,24,17,25,31,38)(14,19,22,32,21,23,36,37)(16,29,28,42,30,26,40,39)(20,35,34,44,27,33,41,43);(5,8,10,9)(6,11,7,12)(15,25)(19,23)(24,38)(26,29)(32,37)(33,35)(39,42)(43,44);(5,9,10,8)(6,11,7,12)(13,16,18,28,17,30,31,40)(14,20,22,34,21,27,36,41)(15,26,24,39,25,29,38,42)(19,33,32,43,23,35,37,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=f,e^2=fh,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, 6718624);