#8504752

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3874 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411948);