#8696143

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	#56267	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603339);