#10069013

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

4120 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=i,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9976209);