#459485

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=i,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366681);