#10368324

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

4473 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:

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

Polycyclic presentation:

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

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, 10275520);