#9923671

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

4210 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)(18,23)(19,26)(21,29)(28,31)(33,34)(35,41)(36,45)(37,39)(38,42)(40,43)(44,46)(47,48);(1,3,7,11)(2,8,9,15)(4,12,14,5)(6,10,13,16)(17,18,22,23)(19,20,26,24)(21,25,29,30)(27,28,32,31)(33,35,40,37)(34,41,43,39)(36,44,38,47)(42,48,45,46);(2,9)(3,11)(4,14)(10,16)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,36,40,38)(34,42,43,45)(35,44,37,47)(39,46,41,48);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,13)(15,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,37,40,35)(34,39,43,41)(36,44,38,47)(42,48,45,46);(1,5,7,12)(2,10,9,16)(3,4,11,14)(6,8,13,15)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,31)(30,32)(33,38,40,36)(34,42,43,45)(35,44,37,47)(39,48,41,46);(1,6)(2,4)(3,10)(5,15)(7,13)(8,12)(9,14)(11,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,39)(34,35)(36,46)(37,43)(38,48)(40,41)(42,44)(45,47)

Polycyclic presentation:

2^9|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, 9830867);