#8533275

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

4185 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440471);