#9947442

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	#57020	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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:

(1,2,6,9)(3,7,11,5)(4,8,14,16)(10,13,12,15)(17,18,22,25)(19,23,27,20)(21,29,31,24)(26,28,32,30)(33,34)(35,42)(36,39)(37,40)(38,44)(41,46)(43,47)(45,48);(1,3,6,11)(2,5,9,7)(4,12,14,10)(8,13,16,15)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,31)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(17,19,22,27)(18,20,25,23)(21,26,31,32)(24,28,29,30)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,29)(31,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4)(2,8)(3,10)(5,15)(6,14)(7,13)(9,16)(11,12)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,37,39,35)(34,40,36,42)(38,43,46,45)(41,48,44,47);(1,5,6,7)(2,3,9,11)(4,13,14,15)(8,12,16,10)(17,21,22,31)(18,24,25,29)(19,26,27,32)(20,28,23,30)(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=h,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, 9854638);