#8093491

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	#52693	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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)(3,8)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(17,18,23,20)(19,26,22,24)(21,28,30,25)(27,31,29,32)(33,34,39,43)(35,42,38,40)(36,37,47,41)(44,45,46,48);(3,11)(4,13)(8,14)(9,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,39,38)(34,40,43,42)(36,44,47,46)(37,48,41,45);(2,7)(3,11)(9,16)(10,12)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,36)(34,41)(35,44)(37,43)(38,46)(39,47)(40,45)(42,48);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,39,41)(34,36,43,47)(35,45,38,48)(40,46,42,44);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(33,35,39,38)(34,42,43,40)(36,44,47,46)(37,45,41,48);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,38,39,35)(34,40,43,42)(36,46,47,44)(37,48,41,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000687);