#7143107

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42648	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1979 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)(7,11)(10,12)(13,14,18,22)(15,23,27,33)(16,20,28,35)(17,30,31,21)(19,25,34,26)(24,37,39,43)(29,42,41,36)(32,38,44,40);(6,9)(7,11)(13,15,18,27)(14,19,22,34)(16,24,28,39)(17,25,31,26)(20,32,35,44)(21,33,30,23)(29,38,41,40)(36,43,42,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,24)(17,29)(18,28)(19,32)(21,36)(22,35)(23,37)(25,38)(26,40)(27,39)(30,42)(31,41)(33,43)(34,44);(6,9)(10,12)(15,25)(17,31)(19,33)(21,30)(23,34)(24,38)(26,27)(29,41)(32,43)(36,42)(37,44)(39,40);(5,7,8,11)(6,10,9,12)(16,28)(20,35)(24,39)(29,41)(32,44)(36,42)(37,43)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050303);