#9913306

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

4233 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=g,f^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, 9820502);