#8158999

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066195);