#8617759

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

4149 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8524955);