#9913024

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

4069 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820220);