#9050401

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	#48418	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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)(3,7)(4,11)(5,13)(6,9)(8,15)(10,12)(14,16)(17,18,23,28)(19,26,21,24)(20,30,29,25)(22,27,31,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3,6,11)(2,7,9,4)(5,10,15,16)(8,14,13,12)(17,19,23,21)(18,24,28,26)(20,31,29,22)(25,32,30,27)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(1,4,6,7)(2,3,9,11)(5,14,15,12)(8,16,13,10)(17,20)(18,25)(19,22)(21,31)(23,29)(24,27)(26,32)(28,30)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,16)(14,15)(17,21,23,19)(18,26,28,24)(20,31,29,22)(25,32,30,27)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,2)(3,7)(4,11)(5,8)(6,9)(10,14)(12,16)(13,15)(17,22,23,31)(18,27,28,32)(19,29,21,20)(24,30,26,25)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,6,15)(2,8,9,13)(3,10,11,16)(4,12,7,14)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8957597);