#6583722

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	#24784	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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:

(5,6,8,9)(7,10,11,12)(13,14)(15,20)(16,28)(17,22)(18,32)(19,23)(21,30)(24,37)(25,34)(26,35)(27,41)(29,36)(31,42)(33,40)(38,44)(39,43);(1,2)(3,4)(6,9)(10,12)(13,15,19,25)(14,20,23,34)(16,24,31,38)(17,26,18,27)(21,33,36,43)(22,35,32,41)(28,37,42,44)(29,39,30,40);(2,4)(13,16,17,29,19,31,18,30)(14,21,32,42,23,36,22,28)(15,24,26,39,25,38,27,40)(20,33,41,44,34,43,35,37);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(15,25)(20,34)(24,38)(26,27)(33,43)(35,41)(37,44)(39,40);(2,4)(13,17,19,18)(14,22,23,32)(15,26,25,27)(16,29,31,30)(20,35,34,41)(21,28,36,42)(24,39,38,40)(33,37,43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490918);