#8127297

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	#59308	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8034493);