#7062884

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×#3309	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1892 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)(15,25)(16,22)(17,23)(18,20)(19,24)(21,26)(27,38)(28,36)(29,35)(30,37)(31,40)(32,39)(33,34)(41,43)(42,44);(5,6)(7,11)(8,10)(9,12)(14,20)(15,26)(16,29)(17,18)(21,36)(23,24)(25,34)(27,41)(30,42)(31,32)(35,40)(37,38);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,30)(17,33)(18,28)(19,26)(20,34)(22,38)(23,36)(24,25)(27,32)(29,41)(31,42)(35,44)(37,39)(40,43);(7,12)(9,11)(13,16)(14,22)(15,27)(17,31)(18,29)(19,32)(20,35)(21,37)(23,40)(24,39)(25,38)(26,30)(28,42)(33,41)(34,43)(36,44);(6,10)(9,11)(13,17,19,18)(14,20,24,23)(15,28,26,33)(16,31,32,29)(21,36,25,34)(22,35,39,40)(27,42,30,41)(37,44,38,43)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,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=dh,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, 6970080);