#9230859

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9138055);