#9224306

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4179 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=gi,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, 9131502);