#379437

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

794 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,18,24,30,22,28,23,29)(19,31,34,42,33,25,21,27)(20,35,39,45,38,26,40,44)(32,41,36,43,47,46,37,48);(1,3,5,4)(2,6,7,8)(17,19,24,34,22,33,23,21)(18,25,30,27,28,31,29,42)(20,36,40,32,38,37,39,47)(26,43,45,41,35,48,44,46);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,14,15,16)(17,20)(18,26)(19,32)(21,37)(22,38)(23,40)(24,39)(25,41)(27,43)(28,35)(29,45)(30,44)(31,46)(33,47)(34,36)(42,48);(2,7)(6,8)(9,12,13,11)(10,14,15,16)(17,21,23,33,22,34,24,19)(18,27,29,25,28,42,30,31)(20,37,40,47,38,36,39,32)(26,43,45,41,35,48,44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=ef,d^2=fh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286633);