#9730359

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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)(4,5)(7,8)(9,13)(10,16)(11,14)(12,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,7,10)(2,6,8,16)(4,11,13,15)(5,14,9,12)(19,25)(22,29)(26,31)(30,32)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,8)(3,10)(4,13)(12,14)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,34,37,40)(35,38,43,41)(36,39,45,48)(42,46,44,47);(1,4)(2,9)(3,11)(5,8)(6,12)(7,13)(10,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,5,7,9)(2,4,8,13)(3,12,10,14)(6,15,16,11)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,6)(2,10)(3,8)(4,12)(5,11)(7,16)(9,15)(13,14)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,e^2=gi,f^2=ghi,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, 9637555);