#8069363

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

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

8533 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976559);