#392524

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

814 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,c^b=ci,d^b=di

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, 299720);