#7699563

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

7298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606759);