#6811429

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
prime radical	#42648	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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)(15,20)(16,21)(17,31)(18,23)(19,24)(22,32)(25,34)(26,35)(27,36)(28,37)(29,40)(30,38)(33,39)(41,43)(42,44);(5,7,10,6)(8,12,9,11)(13,15,19,26)(14,20,24,35)(16,27,28,29)(17,18,32,33)(21,36,37,40)(22,39,31,23)(25,41,42,30)(34,43,44,38);(2,4)(13,16,18,30,19,28,33,41)(14,21,23,38,24,37,39,43)(15,25,17,27,26,42,32,29)(20,34,31,36,35,44,22,40);(5,8,10,9)(6,11,7,12)(16,28)(21,37)(25,42)(27,29)(30,41)(34,44)(36,40)(38,43);(5,9,10,8)(6,11,7,12)(13,17,18,26,19,32,33,15)(14,22,23,20,24,31,39,35)(16,29,30,25,28,27,41,42)(21,36,38,44,37,40,43,34)

Polycyclic presentation:

2^9|b^2=fi,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, 6718625);