#8505033

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	#42462	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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34)(35,37)(36,44)(38,40)(39,41)(42,47)(43,48)(45,46);(2,4)(7,9)(8,12)(15,16)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(33,35,39,43)(34,37,41,48)(36,45,38,42)(40,47,44,46);(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,36,39,38)(34,40,41,44)(35,42,43,45)(37,46,48,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,35)(34,37)(36,42)(38,45)(39,43)(40,46)(41,48)(44,47);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,37)(34,35)(36,46)(38,47)(39,48)(40,42)(41,43)(44,45);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)(33,38,39,36)(34,40,41,44)(35,42,43,45)(37,47,48,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412229);