#8696014

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	#53393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,37,40)(35,41,43,38)(36,39,46,45)(42,48,44,47);(2,3)(7,9)(8,10)(15,16)(18,19)(23,25)(24,26)(30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,48)(45,46);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,48)(45,46);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35,37,43)(34,38,40,41)(36,44,46,42)(39,48,45,47);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,34)(35,38)(36,45)(37,40)(39,46)(41,43)(42,48)(44,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36,37,46)(34,39,40,45)(35,42,43,44)(38,47,41,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=g,e^2=gh,f^2=g,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, 8603210);