#10196248

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4643 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26)(27,31)(28,29)(30,32)(33,37)(34,35)(36,38)(39,40);(1,3,7,6)(2,4,8,5)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27,30,36)(26,31,32,38)(28,34,33,39)(29,35,37,40);(1,4,7,5)(2,3,8,6)(10,15)(11,18)(16,23)(19,20)(26,32)(28,33)(31,38)(34,39);(2,8)(4,5)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,28,30,33)(26,29,32,37)(27,34,36,39)(31,35,38,40);(1,5,7,4)(2,3,8,6)(9,12,14,22)(10,16,15,23)(11,19,18,20)(13,21,17,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,38)(33,39)(37,40);(1,6,7,3)(2,5,8,4)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,29)(26,33)(27,35)(28,32)(30,37)(31,39)(34,38)(36,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=gh,e^2=g,f^2=gi,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, 10103444);