#7145757

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2150 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)(13,14,18,22,19,23,31,17)(15,24,26,34,27,37,32,20)(16,21,30,36,29,35,41,28)(25,38,40,43,39,44,42,33);(6,9)(7,11)(13,15,19,27)(14,20,23,34)(16,25,29,39)(17,24,22,37)(18,32,31,26)(21,33,35,43)(28,38,36,44)(30,42,41,40);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,25)(17,28)(18,30)(19,29)(20,33)(22,36)(23,35)(24,38)(26,40)(27,39)(31,41)(32,42)(34,43)(37,44);(6,9)(10,12)(13,17,31,23,19,22,18,14)(15,20,32,37,27,34,26,24)(16,28,41,35,29,36,30,21)(25,33,42,44,39,43,40,38);(5,7,8,11)(6,10,9,12)(16,29)(21,35)(25,39)(28,36)(30,41)(33,43)(38,44)(40,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7052953);