#8720700

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	#59401	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,d^2=gi,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, 8627896);