#9865957

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

4106 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,23,20)(19,21,24,26)(22,28,31,25)(27,30,32,29)(33,34,39,44)(35,38,43,40)(36,41,37,42)(45,46,47,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,35,39,43)(34,40,44,38)(36,45,37,47)(41,48,42,46);(2,7)(6,13)(8,11)(14,15)(18,20)(19,24)(25,28)(27,32)(33,36,39,37)(34,41,44,42)(35,45,43,47)(38,46,40,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32)(33,37,39,36)(34,42,44,41)(35,45,43,47)(38,46,40,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,38,39,40)(34,43,44,35)(36,46,37,48)(41,47,42,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=i,e^2=gi,f^2=g,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, 9773153);