#10451592

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

4341 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)(4,8)(5,6)(9,10,14,18)(11,19,20,15)(12,16,17,13)(21,22,24,23)(25,26,30,34)(27,35,38,31)(28,39,37,32)(29,40,36,33);(1,3,5,8)(2,4,6,7)(11,20)(15,19)(21,24)(22,23)(25,26)(27,31)(28,32)(29,40)(30,34)(33,36)(35,38)(37,39);(9,11)(10,15)(12,21)(13,22)(14,20)(16,23)(17,24)(18,19)(25,27)(26,31)(28,36)(29,37)(30,38)(32,33)(34,35)(39,40);(2,6)(4,7)(9,12)(10,16)(11,21)(13,18)(14,17)(15,23)(19,22)(20,24)(25,28,30,37)(26,32,34,39)(27,36,38,29)(31,33,35,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,23)(17,24)(18,19)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,32,35,39);(1,2)(3,4)(5,6)(7,8)(9,13,14,16)(10,17,18,12)(11,22,20,23)(15,24,19,21)(25,26)(27,31)(28,39)(29,33)(30,34)(32,37)(35,38)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358788);