#9851054

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

3946 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)(3,10)(4,14)(5,12)(6,8)(7,13)(9,15)(11,16)(17,18)(19,24)(20,25)(21,29)(22,23)(26,31)(27,28)(30,32)(33,34,40,39)(35,44,42,37)(36,41,46,45)(38,43,47,48);(1,3)(2,7)(4,12)(5,9)(6,13)(8,10)(11,15)(14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,32)(29,31)(33,35)(34,37)(36,43)(38,41)(39,44)(40,42)(45,47)(46,48);(2,8)(4,15)(7,10)(11,12)(18,23)(19,25)(26,29)(27,32)(33,36,40,46)(34,41,39,45)(35,43,42,48)(37,38,44,47);(1,4)(2,9)(3,12)(5,7)(6,15)(8,14)(10,16)(11,13)(17,19,22,25)(18,24,23,20)(21,27,31,32)(26,30,29,28)(33,37,40,44)(34,42,39,35)(36,38,46,47)(41,48,45,43);(1,3)(2,10)(4,12)(5,14)(6,13)(7,8)(9,16)(11,15)(17,20,22,24)(18,25,23,19)(21,30,31,28)(26,27,29,32)(33,38)(34,43)(35,45)(36,44)(37,46)(39,48)(40,47)(41,42);(1,5,6,16)(2,11,8,12)(3,9,13,14)(4,7,15,10)(17,21)(18,26)(19,27)(20,28)(22,31)(23,29)(24,30)(25,32)(33,39,40,34)(35,44,42,37)(36,45,46,41)(38,43,47,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=gh,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, 9758250);