#435703

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

785 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342899);