#480185

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

759 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)(17,18,21,25,22,26,35,40,23,27,20,31,33,41,34,44)(19,24,28,38,30,39,42,48,29,37,32,36,43,46,45,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,32)(21,28)(22,30)(23,29)(25,38)(26,39)(27,37)(31,36)(33,43)(34,45)(35,42)(40,48)(41,46)(44,47);(17,20,22,34,23,21,33,35)(18,25,26,40,27,31,41,44)(19,28,30,42,29,32,43,45)(24,36,39,47,37,38,46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,29)(24,37)(28,32)(30,43)(36,38)(39,46)(42,45)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387381);