#10181191

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

3991 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10088387);