#9221172

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	#44151	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=gh,f^2=g,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, 9128368);