#392544

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

818 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,23,27,21,36)(19,24,31,39,29,38,30,45)(20,32,34,40,35,46,33,25)(28,41,44,47,42,48,43,37);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,41)(33,43)(34,44)(35,42)(36,45)(40,47)(46,48);(1,3,5,4)(2,7,6,8)(17,20,21,33,23,35,22,34)(18,25,26,32,27,40,36,46)(19,28,30,43,29,42,31,44)(24,37,39,41,38,47,45,48);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(19,29)(24,38)(28,42)(30,31)(37,47)(39,45)(41,48)(43,44)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ef,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,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, 299740);