#205165

order	512
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1064 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112361);