#8712354

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	#38509	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=h,e^2=g,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, 8619550);