#8987285

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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)(3,4,8,7)(5,9,13,16)(11,12,15,14)(17,18,22,25)(19,23,26,20)(21,24,29,31)(27,30,32,28)(33,34)(35,37)(36,41)(38,48)(39,42)(40,46)(43,45)(44,47);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,35,39,38)(34,37,42,48)(36,44,46,43)(40,45,41,47);(3,8)(4,7)(5,13)(9,16)(19,26)(20,23)(21,29)(24,31)(33,36,39,46)(34,40,42,41)(35,43,38,44)(37,47,48,45);(1,3)(2,7)(4,10)(5,11)(6,8)(9,14)(12,16)(13,15)(17,20,22,23)(18,19,25,26)(21,30,29,28)(24,32,31,27)(33,36)(34,41)(35,44)(37,47)(38,43)(39,46)(40,42)(45,48);(1,4,6,7)(2,8,10,3)(5,12,13,14)(9,15,16,11)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,35)(36,45)(38,42)(39,48)(40,44)(41,43)(46,47);(1,5,6,13)(2,9,10,16)(3,11,8,15)(4,12,7,14)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,38,39,35)(34,37,42,48)(36,43,46,44)(40,45,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8894481);