#8741698

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

4030 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8648894);