#9730325

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3887 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)(3,5)(4,13)(6,15)(7,8)(9,14)(10,12)(11,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34)(35,41)(36,42)(37,47)(38,39)(40,45)(43,48)(44,46);(1,3,7,11)(2,5,8,16)(4,12,14,6)(9,15,13,10)(19,25)(22,29)(26,31)(30,32)(34,39)(36,46)(40,41)(43,47);(2,8)(3,11)(4,14)(10,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35,38,45)(34,40,39,41)(36,43,46,47)(37,44,48,42);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,13)(10,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35)(34,41)(36,43)(37,44)(38,45)(39,40)(42,48)(46,47);(1,5)(2,3)(4,15)(6,13)(7,16)(8,11)(9,12)(10,14)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,36,38,46)(34,42,39,44)(35,43,45,47)(37,41,48,40);(1,6,7,12)(2,10,8,15)(3,4,11,14)(5,9,16,13)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,37,38,48)(34,43,39,47)(35,44,45,42)(36,40,46,41)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,e^2=g,f^2=gi,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, 9637521);