#8746318

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18,22,26)(19,23,29,32)(20,30,28,24)(21,25,27,31)(33,34,39,43)(35,42,37,40)(36,41,47,38)(44,48,46,45);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19,22,29)(18,23,26,32)(20,21,28,27)(24,31,30,25)(33,35,39,37)(34,40,43,42)(36,46,47,44)(38,45,41,48);(3,12)(7,8)(13,16)(14,15)(17,20,22,28)(18,24,26,30)(19,27,29,21)(23,25,32,31)(33,35,39,37)(34,40,43,42)(36,46,47,44)(38,45,41,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,25)(26,30)(31,32)(33,36,39,47)(34,41,43,38)(35,44,37,46)(40,45,42,48);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,27)(28,30)(29,32)(33,37,39,35)(34,42,43,40)(36,46,47,44)(38,45,41,48);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,21,22,27)(18,25,26,31)(19,28,29,20)(23,24,32,30)(33,38,39,41)(34,36,43,47)(35,45,37,48)(40,46,42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,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, 8653514);