#8701366

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	#52294	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4132 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)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18,24,23)(19,22,28,27)(20,29,21,25)(26,32,30,31);(2,3)(7,9)(8,10)(15,16)(17,19)(18,22)(20,30)(21,26)(23,27)(24,28)(25,32)(29,31)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(17,20,24,21)(18,25,23,29)(19,26,28,30)(22,31,27,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,21,24,20)(18,25,23,29)(19,26,28,30)(22,32,27,31)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,22)(18,19)(20,31)(21,32)(23,28)(24,27)(25,26)(29,30)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,23,24,18)(19,27,28,22)(20,29,21,25)(26,32,30,31)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gi,e^2=g,f^2=gi,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, 8608562);