#231725

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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

537 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138921);