#8696733

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	#39140	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 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)(19,24)(20,26)(21,30)(22,28)(23,29)(25,31)(27,32)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(2,3)(7,9)(8,10)(14,15)(17,19,23,22)(18,24,29,28)(20,27,31,30)(21,26,32,25)(35,44)(36,45)(40,46)(41,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20)(18,25)(19,30)(21,28)(22,27)(23,31)(24,32)(26,29)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,26)(19,27)(21,28)(22,30)(23,31)(24,32)(25,29)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,27)(19,26)(20,28)(22,25)(23,32)(24,31)(29,30)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,19)(18,28,29,24)(20,27,31,30)(21,26,32,25)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=g,e^2=g,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, 8603929);