#432045

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

978 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)(11,15)(12,14)(13,16)(17,18)(19,29)(20,24)(21,26)(22,27)(23,31)(25,33)(28,38)(30,46)(32,44)(34,41)(35,42)(36,43)(37,45)(39,47)(40,48);(1,2,5,6)(3,7,4,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,31)(18,23,26,29)(20,34,35,48)(22,33,38,32)(24,40,42,41)(25,27,44,28)(30,37,47,36)(39,45,46,43);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20,21,35)(18,24,26,42)(19,30,31,47)(22,37,38,36)(23,39,29,46)(25,41,44,40)(27,45,28,43)(32,48,33,34);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,19)(18,25)(20,30)(21,31)(22,32)(23,28)(24,41)(26,44)(27,29)(33,38)(34,36)(35,47)(37,48)(39,43)(40,42)(45,46)

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,d^2=efi,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, 339241);