#8692578

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
prime radical	#58255	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31)(34,39)(40,41)(42,45)(47,48);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,34,38,39)(35,40,36,41)(37,42,46,45)(43,47,44,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,36,38,35)(34,41,39,40)(37,43,46,44)(42,47,45,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8599774);