#8898026

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

4453 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:

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

Polycyclic presentation:

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

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, 8805222);