#7664871

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

7780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(2,4)(5,6,9,8)(7,11,12,10)(14,17)(15,19);(5,7,9,12)(6,10,8,11)(14,17)(18,20);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,14,16,17)(15,18,19,20);(13,15)(14,18)(16,19)(17,20);(2,4)(5,6,9,8)(7,10,12,11);(1,2)(3,4)(5,8,9,6)(7,11,12,10)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572067);