#8662967

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	#41222	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18)(19,23)(20,29)(21,24)(22,26)(25,28)(27,32)(30,31)(33,34,38,42)(35,37,41,39)(36,40,44,47)(43,45,48,46);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19)(18,23)(20,30)(21,27)(22,28)(24,32)(25,26)(29,31)(36,44)(40,47)(43,48)(45,46);(17,20,22,21)(18,24,26,29)(19,27,28,30)(23,31,25,32)(33,35,38,41)(34,39,42,37)(36,43,44,48)(40,46,47,45);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,18)(19,25)(20,24)(21,29)(22,26)(23,28)(27,32)(30,31)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,48)(42,47);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,19)(18,25)(20,27)(21,30)(22,28)(23,26)(24,32)(29,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,21,22,20)(18,24,26,29)(19,27,28,30)(23,32,25,31)(33,37,38,39)(34,41,42,35)(36,45,44,46)(40,48,47,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570163);