#10195997

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

4474 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,e^2=g,f^2=i,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, 10103193);