#6374385

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2857 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)(19,27,30,23)(20,24)(21,25)(22,26)(28,41,44,35)(29,42,46,36)(31,37)(32,38)(33,39)(34,40)(43,48,45,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,36)(26,27)(31,43)(32,44)(33,45)(34,46)(37,47)(38,41)(39,48)(40,42);(2,8)(6,9)(7,16)(14,15)(17,20,22,32)(18,24,26,38)(19,28,30,44)(21,33,34,31)(23,35,27,41)(25,39,40,37)(29,45,46,43)(36,48,42,47);(17,21,22,34)(18,25,26,40)(19,29,30,46)(20,31,32,33)(23,36,27,42)(24,37,38,39)(28,43,44,45)(35,47,41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=i,e^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, 6281581);