#8993725

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	#53964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,11)(4,15)(5,10)(6,7)(8,14)(9,13)(12,16)(17,18)(19,21)(20,25)(22,32)(23,26)(24,30)(27,29)(28,31)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,3,7,14)(2,8,6,11)(4,12,13,5)(9,16,15,10)(17,19,23,22)(18,21,26,32)(20,28,30,27)(24,29,25,31)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(2,6)(4,13)(5,12)(8,11)(17,20,23,30)(18,24,26,25)(19,27,22,28)(21,31,32,29)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,4)(2,9)(3,12)(5,14)(6,15)(7,13)(8,16)(10,11)(17,20)(18,25)(19,28)(21,31)(22,27)(23,30)(24,26)(29,32)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,15,11,9)(17,21)(18,19)(20,29)(22,26)(23,32)(24,28)(25,27)(30,31)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,6,7,2)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,22,23,19)(18,21,26,32)(20,27,30,28)(24,29,25,31)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8900921);