#390919

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

954 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)(3,5)(4,7)(6,8)(10,13)(11,15)(17,18,22,27,23,28,21,26)(19,24,30,38,31,39,36,44)(20,25,35,43,33,41,34,42)(29,37,32,40,45,47,46,48);(1,3)(2,4)(5,8)(6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,32)(21,36)(22,30)(23,31)(25,40)(26,44)(27,38)(28,39)(29,34)(33,46)(35,45)(37,42)(41,48)(43,47);(1,4,6,5)(2,7,8,3)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29,31,45)(21,34)(22,35)(23,33)(24,37,39,47)(26,42)(27,43)(28,41)(30,32,36,46)(38,40,44,48);(1,5,6,4)(2,7,8,3)(10,13)(14,16)(20,33)(25,41)(29,45)(32,46)(34,35)(37,47)(40,48)(42,43)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=egh,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298115);