#9851017

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3873 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)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,20)(19,21,24,26)(22,28,31,25)(27,30,32,29)(33,34,38,42)(35,39,43,47)(36,40,46,45)(37,41,48,44);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(35,43)(36,46)(39,47)(40,45);(2,3)(7,9)(8,10)(14,15)(18,20)(19,24)(25,28)(27,32)(33,35,38,43)(34,39,42,47)(36,44,46,41)(37,45,48,40);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32)(33,36,38,46)(34,40,42,45)(35,44,43,41)(37,47,48,39);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,37)(34,41)(35,45)(36,39)(38,48)(40,43)(42,44)(46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=h,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, 9758213);