#7054774

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2089 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)(13,14,20,17)(15,24,29,26)(16,22,32,30)(18,23,19,34)(21,27,33,28)(25,39,42,40)(31,37,35,38)(36,43,44,41);(5,6)(7,11)(8,10)(9,12)(13,15,20,29)(14,21,17,33)(16,25,32,42)(18,28,19,27)(22,36,30,44)(23,26,34,24)(31,43,35,41)(37,39,38,40);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,25)(17,30)(18,35)(19,31)(20,32)(21,36)(23,38)(24,39)(26,40)(27,43)(28,41)(29,42)(33,44)(34,37);(7,12)(9,11)(13,17,20,14)(15,26,29,24)(16,30,32,22)(18,34,19,23)(21,28,33,27)(25,40,42,39)(31,38,35,37)(36,41,44,43);(6,10)(9,11)(13,18,20,19)(14,23,17,34)(15,27,29,28)(16,31,32,35)(21,24,33,26)(22,37,30,38)(25,41,42,43)(36,40,44,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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=dfi,e^b=ehi,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, 6961970);