#10020815

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

4110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,f^2=g,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, 9928011);