#7699586

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

8141 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)(7,11)(8,12)(9,10)(14,18)(19,20);(2,4)(13,14,17,18)(15,19,16,20);(1,2)(3,4)(6,10)(11,12)(13,15,17,16)(14,19,18,20);(5,6)(7,11)(8,12)(9,10)(13,15,17,16)(14,19,18,20);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,16)(14,19,18,20);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(13,16,17,15)(14,20,18,19)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,e^2=gh,f^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, 7606782);