#8180370

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087566);