#9232803

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

3927 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=ghi,e^2=gh,f^2=ghi,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, 9139999);