#197950

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2)(3,4)(6,9)(7,11)(10,12)(13,14,18,22,19,21,17,15)(16,24,27,20,25,23,26,28);(5,6,7,9,8,10,11,12)(13,15,18,21,19,22,17,14)(16,24,26,20,25,23,27,28);(2,4)(6,10)(9,12)(13,16,19,25)(14,20,21,28)(15,23,22,24)(17,27,18,26);(14,21)(15,22)(20,28)(23,24)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105146);