#7054900

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	C₂×#3267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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=ehi,f^b=fi,d^c=dhi,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, 6962096);