#8720651

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	#47182	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4141 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=i,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, 8627847);