#459844

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

891 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)(3,7)(9,10,13,15)(11,14,12,16)(17,18,22,27)(19,24,32,42)(20,35,21,25)(23,28,39,29)(26,38,45,37)(30,41,47,40)(31,44,36,48)(33,43,34,46);(1,2,4,5)(3,6,7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,36)(21,31)(22,32)(23,34)(25,44)(26,41)(27,42)(28,46)(29,43)(30,37)(33,39)(35,48)(38,47)(40,45);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(17,20,22,21)(18,25,27,35)(19,30,32,47)(23,38,39,37)(24,40,42,41)(26,29,45,28)(31,34,36,33)(43,48,46,44);(3,7)(6,8)(9,12,13,11)(10,16,15,14)(17,21,22,20)(18,26,27,45)(19,31,32,36)(23,38,39,37)(24,41,42,40)(25,29,35,28)(30,34,47,33)(43,48,46,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=ei,d^2=ef,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, 367040);