#8720367

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	#2397	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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)(17,18,22,23)(19,20,26,25)(21,24,31,32)(27,28,29,30)(34,39)(35,44)(36,43)(46,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18,22,23)(19,25,26,20)(21,24,31,32)(27,30,29,28)(33,34)(35,40)(36,42)(37,43)(38,39)(41,44)(45,47)(46,48);(4,13)(8,12)(11,16)(14,15)(18,23)(19,26)(24,32)(27,29)(33,35,38,44)(34,40,39,41)(36,45,43,48)(37,46,42,47);(1,4,6,13)(2,8,10,12)(3,11,9,16)(5,15,7,14)(17,19)(18,20)(21,29)(22,26)(23,25)(24,30)(27,31)(28,32)(33,35)(34,41)(36,45)(37,47)(38,44)(39,40)(42,46)(43,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,15)(12,14)(13,16)(17,20)(18,19)(21,30)(22,25)(23,26)(24,29)(27,32)(28,31)(33,36)(34,42)(35,45)(37,39)(38,43)(40,47)(41,46)(44,48);(1,5)(2,9)(3,10)(4,14)(6,7)(8,11)(12,16)(13,15)(17,21,22,31)(18,24,23,32)(19,27,26,29)(20,28,25,30)(33,37,38,42)(34,43,39,36)(35,46,44,47)(40,48,41,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=ghi,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, 8627563);