#8920732

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8827928);