#8463307

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370503);