#8107685

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4259 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014881);