#6676227

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#29816	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1842 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)(5,6,9,11)(7,12,8,10)(14,19)(16,27)(20,32)(22,30)(23,35)(24,39)(28,41)(40,42);(5,6)(7,10)(8,12)(9,11)(13,14,17,19)(15,23,25,35)(16,21,27,33)(18,30,29,22)(20,31,32,26)(24,38,39,44)(28,36,41,37)(34,42,43,40);(2,4)(13,15,17,25)(14,20,19,32)(16,24,27,39)(18,31,29,26)(21,34,33,43)(22,35,30,23)(28,42,41,40)(36,38,37,44);(5,7,9,8)(6,10,11,12)(13,16)(14,21)(15,24)(17,27)(18,28)(19,33)(20,34)(22,37)(23,38)(25,39)(26,40)(29,41)(30,36)(31,42)(32,43)(35,44);(5,8,9,7)(6,10,11,12)(14,22)(16,27)(18,29)(19,30)(20,35)(21,36)(23,32)(24,39)(26,31)(33,37)(34,38)(43,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6583423);