#9815124

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

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9722320);