#9498389

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	#53994	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34,38,39)(35,43,46,40)(36,41,47,45)(37,44,48,42);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(34,39)(36,47)(40,43)(42,44);(2,6)(3,5)(4,12)(15,16)(18,23)(19,26)(24,29)(27,32)(33,35,38,46)(34,40,39,43)(36,42,47,44)(37,41,48,45);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,35)(34,40)(36,42)(37,45)(38,46)(39,43)(41,48)(44,47);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36)(34,41)(35,44)(37,40)(38,47)(39,45)(42,46)(43,48);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,31)(26,32)(33,37)(34,42)(35,45)(36,40)(38,48)(39,44)(41,46)(43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=gh,e^2=gh,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, 9405585);