#8726718

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	#2657	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gi,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, 8633914);