#331578

order	512
nilpotency class	3
isoclinism	isoclinic class of #331114
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

601 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238774);