#456131

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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

906 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ei,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, 363327);