#7149318

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	#30458	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1904 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)(16,22)(17,23)(18,24)(19,35)(20,25)(21,30)(27,42)(28,41)(29,38)(31,43)(32,39)(33,37)(34,40)(36,44);(6,9)(7,11)(13,15,19,30,20,31,18,29)(14,21,24,26,25,38,35,43)(16,27,32,44,17,33,34,28)(22,36,40,42,23,41,39,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,17)(14,22,25,23)(15,27,31,33)(18,34,19,32)(21,36,38,41)(24,40,35,39)(26,42,43,37)(28,30,44,29);(6,9)(10,12)(13,17,20,16)(14,23,25,22)(15,28,31,44)(18,34,19,32)(21,37,38,42)(24,40,35,39)(26,41,43,36)(27,30,33,29);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,25,35)(15,29,31,30)(16,32,17,34)(21,26,38,43)(22,39,23,40)(27,44,33,28)(36,37,41,42)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^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, 7056514);