#480205

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×#532	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

820 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)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,31)(21,28)(22,30)(24,27)(25,39)(26,36)(29,32)(33,44)(34,42)(35,37)(38,41)(40,48)(43,45)(46,47);(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)(19,28)(23,27)(29,31)(30,42)(35,39)(36,41)(43,44)(46,48)

Polycyclic presentation:

2^9|a^2=f,f^2=h,h^2=i,b^a=be,e^a=eg,c^b=ci,d^b=di,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, 387401);