#9713693

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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)(5,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,25)(20,21)(22,26)(23,32)(24,28)(27,29)(30,31)(33,34,39,43)(35,42,38,40)(36,37,41,48)(44,45,46,47);(1,3)(2,6)(4,10)(5,13)(7,9)(8,14)(11,15)(12,16)(17,19)(18,25)(20,27)(21,29)(22,32)(23,26)(24,30)(28,31)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,47,48,45);(2,4)(3,9)(8,12)(11,13)(17,20,24,22)(18,26,28,21)(19,27,30,32)(23,31,29,25)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(1,4,7,2)(3,10,9,6)(5,12,15,8)(11,14,13,16)(17,21,24,26)(18,20,28,22)(19,29,30,23)(25,27,31,32)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,5,7,15)(2,8,4,12)(3,11,9,13)(6,16,10,14)(17,22,24,20)(18,26,28,21)(19,27,30,32)(23,25,29,31)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,6,7,10)(2,9,4,3)(5,14,15,16)(8,11,12,13)(17,23,24,29)(18,27,28,32)(19,26,30,21)(20,31,22,25)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,45,48,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9620889);