#9851005

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

3982 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,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, 9758201);