#7054836

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	C₂×#3283	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2105 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)(13,14,20,24)(15,25,29,37)(16,22,32,38)(17,18,23,19)(21,35,27,28)(26,39,42,44)(30,31,33,34)(36,40,43,41);(5,6)(7,11)(8,10)(9,12)(13,15,20,29)(14,21,24,27)(16,26,32,42)(17,25,23,37)(18,28,19,35)(22,36,38,43)(30,44,33,39)(31,40,34,41);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,26)(17,33)(18,34)(19,31)(20,32)(21,36)(23,30)(24,38)(25,39)(27,43)(28,41)(29,42)(35,40)(37,44);(7,12)(9,11)(13,17)(14,18)(15,27)(16,30)(19,24)(20,23)(21,29)(22,31)(25,28)(26,36)(32,33)(34,38)(35,37)(39,40)(41,44)(42,43);(6,10)(9,11)(13,18,20,19)(14,23,24,17)(15,28,29,35)(16,31,32,34)(21,37,27,25)(22,33,38,30)(26,40,42,41)(36,39,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,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, 6962032);