#8118838

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	#52451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026034);