#432404

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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

791 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ef,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=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, 339600);