#8633867

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	#47508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=hi,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, 8541063);