#9730433

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,38,39)(35,41,46,40)(36,42,47,45)(37,44,48,43);(1,3,7,11)(2,8,9,16)(4,5,14,15)(6,12,13,10)(19,25)(22,29)(26,31)(30,32)(34,39)(36,47)(40,41)(43,44);(2,9)(3,11)(4,14)(10,12)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35,38,46)(34,40,39,41)(36,44,47,43)(37,45,48,42);(1,4)(2,6)(3,5)(7,14)(8,12)(9,13)(10,16)(11,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35)(34,41)(36,44)(37,45)(38,46)(39,40)(42,48)(43,47);(1,5)(2,10)(3,4)(6,16)(7,15)(8,13)(9,12)(11,14)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,36)(34,42)(35,44)(37,40)(38,47)(39,45)(41,48)(43,46);(1,6)(2,4)(3,12)(5,8)(7,13)(9,14)(10,11)(15,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,37,38,48)(34,43,39,44)(35,45,46,42)(36,41,47,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,e^2=ghi,f^2=g,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, 9637629);