#6476845

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469983
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1755 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,22,26,35,41)(19,27,29,37,30,40,34,23)(20,31,33,47,32,24,38,39)(28,43,45,36,44,42,48,46);(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,26,37)(20,28,32,44)(21,34,35,29)(24,36,31,46)(25,40,41,27)(33,48,38,45)(39,43,47,42);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20,22,32)(18,24,26,31)(19,28,30,44)(21,33,35,38)(23,36,37,46)(25,39,41,47)(27,42,40,43)(29,45,34,48);(11,15)(13,16);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(20,32)(24,31)(28,44)(33,38)(36,46)(39,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6384041);