#7149341

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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1873 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)(14,19)(15,24)(17,20)(21,34)(22,27)(23,32)(25,39)(28,30)(31,36)(35,43)(37,38)(40,44);(6,9)(7,11)(13,14,20,32,18,23,17,19)(15,21,33,34,24,36,26,31)(16,27,30,38,29,37,28,22)(25,40,42,44,39,43,41,35);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,21,23,36)(16,25,29,39)(17,26,20,33)(19,31,32,34)(22,35,38,44)(27,40,37,43)(28,41,30,42);(6,9)(10,12)(13,16,18,29)(14,22,23,38)(15,25,24,39)(17,30,20,28)(19,27,32,37)(21,35,36,44)(26,42,33,41)(31,40,34,43);(5,7,8,11)(6,10,9,12)(15,24)(21,36)(25,39)(26,33)(31,34)(35,44)(40,43)(41,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7056537);