#9039142

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	#37245	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(3,8)(4,12)(5,10)(6,7)(9,13)(11,16)(14,15)(17,18,23,27)(19,26,31,24)(20,22,21,25)(28,32,29,30)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,3,7,5)(2,8,6,10)(4,11,13,14)(9,15,12,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(34,39)(35,42)(41,44)(43,48);(2,6)(3,5)(9,12)(11,14)(17,20,23,21)(18,25,27,22)(19,28,31,29)(24,32,26,30)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,7,13)(2,9,6,12)(3,11,5,14)(8,15,10,16)(17,21,23,20)(18,25,27,22)(19,29,31,28)(24,32,26,30)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,5,7,3)(2,10,6,8)(4,11,13,14)(9,15,12,16)(17,19)(18,26)(20,29)(21,28)(22,30)(23,31)(24,27)(25,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,6,7,2)(3,10,5,8)(4,12,13,9)(11,16,14,15)(17,22,23,25)(18,20,27,21)(19,30,31,32)(24,28,26,29)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8946338);