#480224

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#121	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

747 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,20,25,22,26,33,40,21,24,32,37,34,38,45,47)(19,27,29,39,30,41,43,48,28,23,31,35,42,36,44,46);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,28)(18,23,24,27)(20,31,32,29)(22,30,34,42)(25,39,37,35)(26,36,38,41)(33,44,45,43)(40,48,47,46);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(19,28)(23,27)(29,31)(30,42)(35,39)(36,41)(43,44)(46,48);(4,12)(7,14)(10,16)(13,15)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,c^b=ci,f^b=fg,d^c=dgi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387420);