#8107519

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

4356 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=ghi,e^2=h,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, 8014715);