#7148497

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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)(7,11)(10,12)(13,14)(15,26)(16,23)(17,22)(18,24)(19,35)(20,25)(21,30)(27,37)(28,36)(29,38)(31,42)(32,43)(33,41)(34,40)(39,44);(6,9)(7,11)(13,15,20,31)(14,21,25,38)(16,32,17,27)(18,30,19,29)(22,39,23,36)(24,42,35,26)(28,33,44,34)(37,41,43,40);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,27)(17,20)(18,34)(19,33)(21,36)(23,25)(24,41)(26,43)(28,29)(30,44)(31,32)(35,40)(37,42)(38,39);(6,9)(10,12)(13,17,20,16)(14,23,25,22)(15,28,31,44)(18,34,19,33)(21,37,38,43)(24,41,35,40)(26,39,42,36)(27,30,32,29);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,25,35)(15,29,31,30)(16,33,17,34)(21,26,38,42)(22,40,23,41)(27,44,32,28)(36,37,39,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,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, 7055693);