#7054634

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#24435	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2128 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)(15,21)(16,22)(17,24)(18,23)(19,25)(20,27)(26,36)(28,37)(29,44)(30,41)(31,40)(32,35)(33,39)(34,42)(38,43);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,20,25,35)(16,29,31,42)(17,27,18,32)(21,24,37,23)(22,38,40,36)(26,33,43,30)(34,41,44,39);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,26)(17,32)(18,27)(19,28)(20,23)(22,36)(24,35)(25,37)(29,30)(31,43)(33,42)(34,39)(38,40)(41,44);(7,12)(9,11)(13,16,19,31)(14,22,25,40)(15,26,28,43)(17,30,18,33)(20,34,35,44)(21,36,37,38)(23,39,24,41)(27,42,32,29);(6,10)(9,11)(13,17,19,18)(14,23,25,24)(15,27,28,32)(16,30,31,33)(20,21,35,37)(22,39,40,41)(26,42,43,29)(34,36,44,38)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=i,e^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=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961830);