#7149324

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

1889 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)(15,23)(16,20)(17,31)(18,21)(19,26)(22,24)(25,37)(27,38)(28,43)(29,35)(30,44)(32,42)(33,39)(34,41)(36,40);(6,9)(7,11)(13,15,24,39,18,27,17,26)(14,19,31,38,21,33,22,23)(16,28,40,42,29,25,30,41)(20,34,44,37,35,32,36,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,29)(14,20,21,35)(15,25,27,28)(17,30,24,40)(19,32,33,34)(22,36,31,44)(23,37,38,43)(26,42,39,41);(6,9)(10,12)(15,26)(16,29)(17,24)(19,23)(20,35)(22,31)(25,41)(27,39)(28,42)(32,43)(33,38)(34,37);(5,7,8,11)(6,10,9,12)(14,21)(15,27)(16,29)(22,31)(26,39)(30,40)(32,34)(37,43)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,e^2=hi,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, 7056520);