#7054710

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×#3285	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2109 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)(15,24,28,26)(16,22,31,29)(18,23,19,33)(21,35,32,27)(25,39,43,40)(30,37,34,38)(36,41,44,42);(5,6)(7,11)(8,10)(9,12)(13,15,20,28)(14,21,17,32)(16,25,31,43)(18,27,19,35)(22,36,29,44)(23,26,33,24)(30,41,34,42)(37,39,38,40);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,25)(17,29)(18,34)(19,30)(20,31)(21,36)(23,38)(24,39)(26,40)(27,42)(28,43)(32,44)(33,37)(35,41);(7,12)(9,11)(13,17,20,14)(15,26,28,24)(16,29,31,22)(18,33,19,23)(21,27,32,35)(25,40,43,39)(30,38,34,37)(36,42,44,41);(6,10)(9,11)(13,18,20,19)(14,23,17,33)(15,27,28,35)(16,30,31,34)(21,26,32,24)(22,37,29,38)(25,41,43,42)(36,39,44,40)

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=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, 6961906);