#9865933

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
prime radical	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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)(34,40)(36,44)(41,46)(42,45);(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)(33,34)(35,41)(36,38)(37,46)(39,40)(42,43)(44,48)(45,47);(2,7)(6,13)(8,11)(14,15)(18,20)(19,24)(25,28)(27,32)(33,35,39,37)(34,41,40,46)(36,42,44,45)(38,43,48,47);(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)(33,36)(34,38)(35,42)(37,45)(39,44)(40,48)(41,43)(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)(33,37,39,35)(34,41,40,46)(36,45,44,42)(38,43,48,47);(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)(34,36)(35,43)(37,47)(39,48)(40,44)(41,42)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=i,e^2=gi,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, 9773129);