#510703

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efi,c^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417899);