#6577039

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,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^c=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484235);