#8701354

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	#46054	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4187 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)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32);(2,3)(7,9)(8,10)(15,16)(17,18,23,22)(19,26,20,24)(21,25,30,29)(27,32,28,31)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(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)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(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)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(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)(17,22,23,18)(19,26,20,24)(21,25,30,29)(27,31,28,32)(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=i,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, 8608550);