#10171854

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	#55044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=hi,e^2=i,f^2=g,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, 10079050);