#10195456

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

4045 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)(3,11)(4,12)(5,9)(6,14)(7,8)(10,13)(15,16)(17,18,23,20)(19,24,27,28)(21,25,31,30)(22,29,32,26)(34,40)(36,46)(41,43)(44,47);(1,3,7,12)(2,4,8,11)(5,13,15,6)(9,14,16,10)(17,19)(18,24)(20,28)(21,26)(22,25)(23,27)(29,31)(30,32)(33,34,39,40)(35,41,37,43)(36,42,46,38)(44,48,47,45);(2,8)(4,11)(9,16)(10,14)(18,20)(19,27)(25,30)(26,29)(33,35,39,37)(34,41,40,43)(36,44,46,47)(38,45,42,48);(1,4,7,11)(2,3,8,12)(5,14,15,10)(6,9,13,16)(17,20,23,18)(19,28,27,24)(21,25,31,30)(22,29,32,26)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,48)(43,45);(1,5,7,15)(2,9,8,16)(3,6,12,13)(4,10,11,14)(17,21)(18,25)(19,29)(20,30)(22,28)(23,31)(24,32)(26,27)(33,37,39,35)(34,43,40,41)(36,44,46,47)(38,45,42,48);(1,6)(2,10)(3,5)(4,9)(7,13)(8,14)(11,16)(12,15)(17,22)(18,26)(19,25)(20,29)(21,24)(23,32)(27,30)(28,31)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,44,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,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, 10102652);