#9002443

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	#44594	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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)(3,7,10,5)(4,8,12,15)(11,13,16,14)(17,18,22,20)(19,25,27,23)(21,28,30,24)(26,31,29,32)(33,34,39,43)(35,40,38,48)(36,37,42,41)(44,46,45,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,13)(12,16)(14,15)(18,20)(23,25)(24,28)(31,32)(33,35,39,38)(34,40,43,48)(36,45,42,44)(37,47,41,46);(3,10)(4,12)(5,7)(8,15)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,36,39,42)(34,41,43,37)(35,44,38,45)(40,47,48,46);(1,4)(2,8)(3,11)(5,14)(6,12)(7,13)(9,15)(10,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,36)(34,37)(35,45)(38,44)(39,42)(40,47)(41,43)(46,48);(1,5,6,7)(2,3,9,10)(4,13,12,14)(8,16,15,11)(17,20,22,18)(19,23,27,25)(21,28,30,24)(26,31,29,32)(33,37)(34,42)(35,46)(36,43)(38,47)(39,41)(40,45)(44,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,13)(12,16)(14,15)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,38,39,35)(34,40,43,48)(36,44,42,45)(37,47,41,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8909639);