#9874927

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	#55450	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34,39,37)(35,43,46,40)(36,47,45,41)(38,42,44,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18,23,20)(19,25,21,24)(22,26,30,28)(27,32,29,31)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,48)(42,47);(2,7)(6,13)(8,11)(14,15)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,34,39,37)(35,40,46,43)(36,41,45,47)(38,42,44,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,48,43,42);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21,23,19)(18,25,20,24)(22,27,30,29)(26,31,28,32)(33,37,39,34)(35,40,46,43)(36,47,45,41)(38,42,44,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,47,43,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=gi,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, 9782123);