#10368603

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

4673 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=hi,e^2=h,f^2=gh,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, 10275799);