#9039404

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=gi,e^2=gh,f^2=g,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, 8946600);