#8726270

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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)(3,8)(4,12)(5,9)(6,10)(7,13)(11,15)(14,16)(33,34,40,45)(35,41,47,48)(36,44,38,42)(37,43,46,39);(1,3)(2,5)(4,11)(6,9)(7,15)(8,10)(12,14)(13,16)(17,18,22,23)(19,24,26,20)(21,25,31,32)(27,30,29,28)(33,35,40,47)(34,41,45,48)(36,37,38,46)(39,44,43,42);(4,13)(7,12)(11,16)(14,15)(18,23)(19,26)(25,32)(27,29)(33,36,40,38)(34,42,45,44)(35,46,47,37)(39,41,43,48);(1,4)(2,7)(3,11)(5,15)(6,13)(8,14)(9,16)(10,12)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,31)(30,32)(33,37,40,46)(34,43,45,39)(35,36,47,38)(41,44,48,42);(1,3)(2,8)(4,11)(5,10)(6,9)(7,14)(12,15)(13,16)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,29,32,27)(33,38,40,36)(34,44,45,42)(35,46,47,37)(39,41,43,48);(1,5,6,8)(2,9,10,3)(4,14,13,15)(7,11,12,16)(17,21,22,31)(18,25,23,32)(19,27,26,29)(20,28,24,30)(33,39,40,43)(34,37,45,46)(35,44,47,42)(36,41,38,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=i,f^2=gi,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, 8633466);