#7145806

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2162 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)(13,14,20,17,19,24,18,23)(15,25,33,21,28,38,27,36)(16,22,34,32,30,37,31,29)(26,39,44,35,41,43,42,40);(6,9)(7,11)(13,15,19,28)(14,21,24,36)(16,26,30,41)(17,25,23,38)(18,33,20,27)(22,35,37,40)(29,43,32,39)(31,44,34,42);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,26)(17,32)(18,31)(19,30)(20,34)(21,35)(23,29)(24,37)(25,39)(27,42)(28,41)(33,44)(36,40)(38,43);(6,9)(10,12)(13,17,20,24,19,23,18,14)(15,21,33,38,28,36,27,25)(16,29,34,22,30,32,31,37)(26,40,44,39,41,35,42,43);(5,7,8,11)(6,10,9,12)(16,30)(22,37)(26,41)(29,32)(31,34)(35,40)(39,43)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7053002);