#8899597

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806793);