#9863038

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

4032 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)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34,39,42)(35,43,45,38)(36,40,37,41)(44,48,47,46);(1,3)(2,7)(4,12)(5,9)(6,13)(8,10)(11,15)(14,16)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,35)(34,38)(36,44)(37,47)(39,45)(40,46)(41,48)(42,43);(2,8)(4,15)(7,10)(11,12)(18,25)(19,30)(20,27)(28,32)(33,36,39,37)(34,40,42,41)(35,44,45,47)(38,46,43,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,14)(10,16)(11,13)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,34)(35,38)(36,40)(37,41)(39,42)(43,45)(44,46)(47,48);(1,3)(2,10)(4,12)(5,14)(6,13)(7,8)(9,16)(11,15)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,37,39,36)(34,41,42,40)(35,44,45,47)(38,46,43,48);(1,5,6,16)(2,11,8,12)(3,9,13,14)(4,7,15,10)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,38)(34,35)(36,46)(37,48)(39,43)(40,44)(41,47)(42,45)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=i,e^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, 9770234);