#8607428

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	#49389	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8514624);