#7149298

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1879 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)(15,24,28,39)(16,22,31,29)(18,23,19,33)(21,26,32,27)(25,38,41,44)(30,36,34,37)(35,42,43,40);(6,9)(7,11)(13,15,19,27,20,28,18,26)(14,21,23,24,17,32,33,39)(16,25,30,40,31,41,34,42)(22,35,37,38,29,43,36,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,25)(17,29)(18,34)(19,30)(20,31)(21,35)(23,37)(24,38)(26,42)(27,40)(28,41)(32,43)(33,36)(39,44);(6,9)(10,12)(13,17,20,14)(15,24,28,39)(16,29,31,22)(18,33,19,23)(21,26,32,27)(25,38,41,44)(30,37,34,36)(35,42,43,40);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,17,33)(15,26,28,27)(16,30,31,34)(21,24,32,39)(22,36,29,37)(25,40,41,42)(35,44,43,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,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=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056494);