#8096179

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	#46070	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4144 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)(3,7,4,8)(5,9,13,16)(11,14,12,15)(17,18)(19,22)(20,24)(21,25)(23,28)(26,30)(27,31)(29,32)(33,34,40,44)(35,39,41,36)(37,45,48,42)(38,43,46,47);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30)(33,35,40,41)(34,36,44,39)(37,46,48,38)(42,47,45,43);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,20)(18,23)(19,26)(21,28)(22,29)(24,25)(27,32)(30,31)(33,36,40,39)(34,41,44,35)(37,43,48,47)(38,42,46,45);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,30)(25,31)(28,32)(33,37)(34,42)(35,38)(36,43)(39,47)(40,48)(41,46)(44,45);(1,4,6,3)(2,8,10,7)(5,11,13,12)(9,14,16,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,25)(27,32)(29,31)(33,38)(34,43)(35,37)(36,42)(39,45)(40,46)(41,48)(44,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,29)(27,31)(30,32)(33,39,40,36)(34,41,44,35)(37,47,48,43)(38,42,46,45)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003375);