#9812971

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	#47878	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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)(4,5)(6,7)(8,12)(10,15)(11,16)(13,14)(17,18)(19,24)(20,22)(21,26)(23,25)(27,29)(28,31)(30,32)(33,34,39,41)(35,43,37,40)(36,46,42,38)(44,45,47,48);(2,7)(4,13)(8,9)(10,16)(17,19,23,20)(18,24,25,22)(21,27,30,28)(26,29,32,31)(33,35,39,37)(34,40,41,43)(36,44,42,47)(38,45,46,48);(18,25)(19,20)(26,32)(27,28)(34,41)(36,42)(40,43)(44,47);(1,3,6,12)(2,8,7,9)(4,10,13,16)(5,11,14,15)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,31,32,29)(33,36,39,42)(34,38,41,46)(35,44,37,47)(40,45,43,48);(1,4,6,13)(2,5,7,14)(3,10,12,16)(8,11,9,15)(17,21)(18,26)(19,27)(20,28)(22,31)(23,30)(24,29)(25,32)(33,37,39,35)(34,40,41,43)(36,47,42,44)(38,45,46,48);(1,5)(2,4)(3,11)(6,14)(7,13)(8,10)(9,16)(12,15)(17,22)(18,19)(20,25)(21,29)(23,24)(26,28)(27,32)(30,31)(33,38)(34,42)(35,45)(36,41)(37,48)(39,46)(40,47)(43,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gi,e^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, 9720167);