#415491

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

816 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=e,c^2=e,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 322687);