#430194

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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

912 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)(9,10,13,15)(11,14,12,16)(17,18,21,26)(19,29,32,23)(20,24,36,43)(22,27,39,28)(25,37,44,38)(30,45,47,40)(31,42,35,48)(33,46,34,41);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19,21,32)(18,23,26,29)(20,35,36,31)(22,34,39,33)(24,42,43,48)(25,45,44,40)(27,46,28,41)(30,38,47,37);(1,4,5,2)(3,7,8,6)(10,15)(14,16)(17,20)(18,24)(19,30)(21,36)(22,38)(23,40)(25,27)(26,43)(28,44)(29,45)(31,33)(32,47)(34,35)(37,39)(41,48)(42,46);(3,8)(6,7)(9,12,13,11)(10,16,15,14)(17,20)(18,25)(19,31)(21,36)(22,37)(23,40)(24,27)(26,44)(28,43)(29,45)(30,33)(32,35)(34,47)(38,39)(41,42)(46,48)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=ei,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=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, 337390);