#8664424

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	#53165	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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571620);