#7655168

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

9670 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)(13,14,16,18)(15,19,20,17);(1,2)(3,4)(5,6,10,9)(7,12,8,11)(13,15,16,20)(14,17,18,19);(5,7,10,8)(6,11,9,12)(13,15)(14,17)(16,20)(18,19);(13,14)(15,17)(16,18)(19,20);(5,8,10,7)(6,11,9,12);(5,9,10,6)(7,12,8,11)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304563, #72304599, #72304606
maximal normal subgroup of	#72304563, #72304599, #72304606

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562364);