#8728723

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

4118 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,26)(21,24)(22,28)(23,25)(29,32)(30,31)(33,34,38,40)(35,39,43,44)(36,45,46,41)(37,42,48,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,25,27)(20,28,30,29)(22,26,32,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,44)(45,48)(46,47);(3,10)(6,7)(11,16)(14,15)(18,25)(20,30)(24,27)(28,29)(34,40)(35,43)(41,45)(42,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,30)(18,26,25,31)(19,29,21,28)(22,27,32,24)(33,34)(35,44)(36,41)(37,47)(38,40)(39,43)(42,48)(45,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,23,19)(18,27,25,24)(20,28,30,29)(22,26,32,31)(33,36)(34,41)(35,42)(37,39)(38,46)(40,45)(43,47)(44,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22,23,32)(18,28,25,29)(19,26,21,31)(20,24,30,27)(33,37,38,48)(34,42,40,47)(35,41,43,45)(36,44,46,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,e^2=g,f^2=h,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, 8635919);