#10195462

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

4059 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10102658);