#9923672

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

4370 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,15)(12,16)(17,18,23,20)(19,27,29,24)(21,25,28,30)(22,31,32,26)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(1,3,7,11)(2,8,9,15)(4,12,14,5)(6,10,13,16)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,9)(3,11)(4,14)(10,16)(18,20)(24,27)(25,30)(26,31)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,13)(15,16)(17,20,23,18)(19,27,29,24)(21,30,28,25)(22,31,32,26)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,5,7,12)(2,10,9,16)(3,4,11,14)(6,8,13,15)(17,21)(18,25)(19,22)(20,30)(23,28)(24,26)(27,31)(29,32)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,6)(2,4)(3,10)(5,15)(7,13)(8,12)(9,14)(11,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=hi,e^2=gh,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, 9830868);