#8746393

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	#42566	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4206 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653589);