#6374637

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	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,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, 6281833);