#9815305

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

4018 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)(17,18,23,28)(19,24,30,21)(20,26,32,25)(22,31,29,27)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,3)(2,7)(4,12)(5,9)(6,13)(8,10)(11,15)(14,16)(17,19,23,30)(18,24,28,21)(20,29,32,22)(25,31,26,27)(35,41)(39,42)(43,45)(47,48);(2,8)(4,15)(7,10)(11,12)(17,20,23,32)(18,25,28,26)(19,22,30,29)(21,31,24,27)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,4)(2,9)(3,12)(5,7)(6,15)(8,14)(10,16)(11,13)(17,20)(18,26)(19,29)(21,31)(22,30)(23,32)(24,27)(25,28)(33,36,38,34)(35,42,41,39)(37,44,46,40)(43,48,45,47);(1,3)(2,10)(4,12)(5,14)(6,13)(7,8)(9,16)(11,15)(17,21)(18,19)(20,31)(22,25)(23,24)(26,29)(27,32)(28,30)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,5,6,16)(2,11,8,12)(3,9,13,14)(4,7,15,10)(17,22,23,29)(18,27,28,31)(19,20,30,32)(21,26,24,25)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=ghi,f^2=h,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, 9722501);