#7149389

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 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)(13,14,20,17)(15,24,28,26)(16,29,32,22)(18,23,19,34)(21,35,33,27)(25,42,30,41)(31,40,36,39)(37,44,38,43);(6,9)(7,11)(13,15,18,35,20,28,19,27)(14,21,34,26,17,33,23,24)(16,30,36,44,32,25,31,43)(22,38,39,41,29,37,40,42);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,32)(14,22,17,29)(15,25,28,30)(18,36,19,31)(21,37,33,38)(23,40,34,39)(24,41,26,42)(27,44,35,43);(6,9)(10,12)(13,17,20,14)(15,26,28,24)(16,22,32,29)(18,34,19,23)(21,27,33,35)(25,41,30,42)(31,39,36,40)(37,43,38,44);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,17,34)(15,27,28,35)(16,31,32,36)(21,26,33,24)(22,39,29,40)(25,43,30,44)(37,41,38,42)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,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, 7056585);