#6810505

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
prime radical	#42643	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1997 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)(7,11)(8,10)(9,12)(13,14)(15,21)(16,22)(17,23)(18,33)(19,25)(20,26)(24,34)(27,35)(28,36)(29,37)(30,38)(31,40)(32,41)(39,43)(42,44);(5,7,8,12)(6,9,10,11)(13,15,20,28)(14,21,26,36)(16,29,32,43)(17,18,19,34)(22,37,41,39)(23,33,25,24)(27,30,42,31)(35,38,44,40);(2,4)(7,12)(9,11)(13,16,19,31,20,32,17,30)(14,22,25,40,26,41,23,38)(15,27,18,29,28,42,34,43)(21,35,33,37,36,44,24,39);(6,10)(9,11)(13,17,20,19)(14,23,26,25)(15,18,28,34)(16,30,32,31)(21,33,36,24)(22,38,41,40)(27,29,42,43)(35,37,44,39);(13,18,19,28,20,34,17,15)(14,24,25,21,26,33,23,36)(16,29,31,42,32,43,30,27)(22,39,40,35,41,37,38,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=fh,d^2=i,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,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, 6717701);