#480230

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	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

735 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,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, 387426);