#6374387

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342674
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,h^c=hi,e^d=ei

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, 6281583);