#9545813

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	#41968	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3851 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9453009);