#6993965

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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2161 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)(7,9)(8,10)(11,12)(13,14,18,23)(15,20,25,34)(16,21,28,36)(17,19,29,22)(24,33,39,43)(26,31,40,35)(27,37,30,32)(38,44,41,42);(5,6)(7,11)(8,10)(9,12)(14,19)(16,27)(17,29)(20,31)(21,36)(22,23)(24,38)(26,40)(28,30)(33,43)(34,35)(39,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,24)(17,26)(18,25)(19,31)(21,33)(22,35)(23,34)(27,38)(28,39)(29,40)(30,41)(32,42)(36,43)(37,44);(7,12)(9,11)(13,16,18,28)(14,21,23,36)(15,24,25,39)(17,30,29,27)(19,32,22,37)(20,33,34,43)(26,41,40,38)(31,42,35,44);(6,10)(9,11)(15,25)(20,34)(24,39)(26,40)(31,35)(33,43)(38,41)(42,44)

Polycyclic presentation:

2^9|a^2=i,d^2=i,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, 6901161);