#9730634

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	#40433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,32)(30,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,22,29)(18,23,26,27)(20,21,28,31)(24,30,32,25)(35,41)(38,45)(42,47)(46,48);(2,4)(3,10)(9,12)(11,16)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20,22,28)(18,24,26,32)(19,21,29,31)(23,30,27,25)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21)(18,25)(19,28)(20,29)(22,31)(23,24)(26,30)(27,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,29)(28,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=g,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, 9637830);