#9863965

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

4135 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)(3,10,5,7)(4,9,12,15)(11,16,13,14)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34,38,41)(35,39,43,36)(37,45,47,40)(42,44,48,46);(1,3,6,5)(2,7,8,10)(4,11,12,13)(9,14,15,16)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,34,38,41)(35,39,43,36)(37,40,47,45)(42,46,48,44);(2,8)(7,10)(9,15)(14,16)(18,25)(19,30)(20,27)(28,32)(33,35,38,43)(34,36,41,39)(37,42,47,48)(40,44,45,46);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,15)(13,16)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,36,38,39)(34,35,41,43)(37,46,47,44)(40,48,45,42);(1,5,6,3)(2,10,8,7)(4,11,12,13)(9,14,15,16)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,37)(34,40)(35,42)(36,44)(38,47)(39,46)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=ghi,e^2=i,f^2=g,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, 9771161);