#9230891

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	#43377	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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)(3,12)(4,9)(5,6)(7,10)(8,11)(13,15)(14,16)(17,18)(19,20)(21,29)(22,25)(23,27)(24,32)(26,30)(28,31)(33,34,38,43)(35,42,45,39)(36,46,47,40)(37,48,44,41);(1,3)(2,7)(4,13)(5,14)(6,15)(8,10)(9,16)(11,12)(17,19,22,27)(18,20,25,23)(21,26,32,31)(24,28,29,30)(33,35,38,45)(34,39,43,42)(36,44,47,37)(40,41,46,48);(1,4,8,5)(2,9,11,6)(3,13,10,14)(7,16,12,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,29)(31,32)(33,36,38,47)(34,40,43,46)(35,44,45,37)(39,41,42,48);(1,5,8,4)(2,9,11,6)(3,14,10,13)(7,16,12,15)(17,20,22,23)(18,19,25,27)(21,30,32,28)(24,31,29,26)(33,37)(34,41)(35,36)(38,44)(39,46)(40,42)(43,48)(45,47);(1,6,8,9)(2,5,11,4)(3,15,10,16)(7,14,12,13)(17,19)(18,20)(21,31)(22,27)(23,25)(24,30)(26,32)(28,29)(33,35)(34,42)(36,44)(37,47)(38,45)(39,43)(40,48)(41,46);(1,7,8,12)(2,10,11,3)(4,16,5,15)(6,13,9,14)(17,21,22,32)(18,24,25,29)(19,26,27,31)(20,28,23,30)(33,34)(35,39)(36,40)(37,48)(38,43)(41,44)(42,45)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,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, 9138087);