#8188193

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(3,7)(4,9)(5,13)(6,8)(10,14)(11,12)(15,16)(17,18,23,22)(19,26,24,20)(21,29,30,25)(27,28,31,32)(33,34,40,39)(35,45,47,41)(36,48,46,42)(37,44,38,43);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,19,23,24)(18,20,22,26)(21,27,30,31)(25,28,29,32)(33,35)(34,41)(36,37)(38,46)(39,45)(40,47)(42,44)(43,48);(2,8)(3,9)(10,13)(11,16)(17,20,23,26)(18,24,22,19)(21,28,30,32)(25,31,29,27)(33,36,40,46)(34,42,39,48)(35,37,47,38)(41,44,45,43);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,37,40,38)(34,43,39,44)(35,36,47,46)(41,48,45,42);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21,23,30)(18,25,22,29)(19,27,24,31)(20,28,26,32)(33,38,40,37)(34,44,39,43)(35,46,47,36)(41,42,45,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,22,23,18)(19,26,24,20)(21,29,30,25)(27,32,31,28)(33,39,40,34)(35,45,47,41)(36,42,46,48)(37,44,38,43)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=gi,e^2=h,f^2=g,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, 8095389);