#9851033

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
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3889 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=h,e^2=gi,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, 9758229);