#8746952

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 8654148);