#478167

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

768 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385363);