#6519530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,5)(3,6,7,8)(17,18,23,28)(19,29,32,25)(20,24,34,39)(21,26,35,41)(22,37,30,27)(31,43,45,40)(33,38,46,48)(36,47,44,42);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,30,34,23,32,22,20)(18,24,37,25,28,39,27,29)(21,31,44,46,35,45,36,33)(26,38,47,40,41,48,42,43);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,22,32,23,34,30,19)(18,25,27,24,28,29,37,39)(21,33,36,45,35,46,44,31)(26,40,42,38,41,43,47,48);(9,11,12,15)(10,14,13,16)(17,21,23,35)(18,26,28,41)(19,31,32,45)(20,33,34,46)(22,36,30,44)(24,38,39,48)(25,40,29,43)(27,42,37,47);(10,13)(14,16)(21,35)(26,41)(31,45)(33,46)(36,44)(38,48)(40,43)(42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6426726);