#380895

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

962 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,27,23,28,21,26)(19,24,32,40,30,38,31,39)(20,25,34,42,35,43,36,44)(29,37,33,41,45,47,46,48);(2,5)(6,8)(17,19,21,31,23,30,22,32)(18,24,26,39,28,38,27,40)(20,33,34,45,35,46,36,29)(25,41,42,47,43,48,44,37);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,37)(26,44)(27,42)(28,43)(30,45)(31,46)(32,33)(38,47)(39,48)(40,41);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(19,30)(24,38)(29,45)(31,32)(33,46)(37,47)(39,40)(41,48)

Polycyclic presentation:

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

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, 288091);