#9947455

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	#56340	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854651);