#6816222

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1749 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)(7,11)(8,10)(9,12)(13,14,19,24)(15,20,27,35)(16,21,25,33)(17,22,28,36)(18,31,32,23)(26,40,41,34)(29,43,39,37)(30,44,42,38);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,28)(17,25)(18,29)(19,27)(21,36)(22,33)(23,37)(24,35)(26,30)(31,43)(32,39)(34,38)(40,44)(41,42);(1,2,3,4)(6,10)(7,12)(13,16,28,15,19,25,17,27)(14,21,36,20,24,33,22,35)(18,29,42,26,32,39,30,41)(23,37,44,34,31,43,38,40);(6,10)(9,11)(13,17,19,28)(14,22,24,36)(15,25,27,16)(18,30,32,42)(20,33,35,21)(23,38,31,44)(26,39,41,29)(34,43,40,37);(13,18,17,30,19,32,28,42)(14,23,22,38,24,31,36,44)(15,26,16,29,27,41,25,39)(20,34,21,37,35,40,33,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,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,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723418);