#411512

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ehi,c^2=e,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^c=fi

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, 318708);