#8720740

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	#43476	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4193 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)(3,5)(4,12)(6,9)(7,11)(8,13)(10,16)(14,15)(17,18,22,23)(19,24,28,27)(20,30,31,25)(21,26,32,29)(33,34)(35,43)(36,38)(37,41)(39,40)(42,46)(44,45)(47,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,16)(12,14)(13,15)(17,18,22,23)(19,27,28,24)(20,25,31,30)(21,26,32,29)(34,40)(36,46)(41,43)(44,47);(4,13)(8,12)(10,15)(14,16)(18,23)(19,28)(25,30)(26,29)(33,35,39,37)(34,41,40,43)(36,47,46,44)(38,45,42,48);(1,4,6,13)(2,8,9,12)(3,10,11,15)(5,14,7,16)(17,19,22,28)(18,24,23,27)(20,29,31,26)(21,25,32,30)(33,36,39,46)(34,42,40,38)(35,44,37,47)(41,45,43,48);(1,5,6,7)(2,3,9,11)(4,14,13,16)(8,10,12,15)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28)(33,37,39,35)(34,43,40,41)(36,47,46,44)(38,45,42,48);(1,3)(2,5)(4,15)(6,11)(7,9)(8,16)(10,13)(12,14)(17,21,22,32)(18,26,23,29)(19,30,28,25)(20,24,31,27)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,47,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627936);