#231734

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	#532	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

531 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(2,6,8,13)(7,9,14,15)(17,18,21,26,22,27,31,25)(19,23,20,29,28,24,30,32);(1,2,4,13,5,8,11,6)(3,9,12,7,10,15,16,14)(17,19,21,20,22,28,31,30)(18,23,26,29,27,24,25,32);(3,10)(7,14)(9,15)(12,16)(17,20,21,28,22,30,31,19)(18,24,25,29,27,23,26,32);(1,3,11,16,5,10,4,12)(2,7,13,15,8,14,6,9)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138930);