#7660657

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

7448 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7567853);