#230411

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#2361
prime radical	#19125	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,3,5,7,8,6)(9,10)(11,22,28,38)(12,13)(14,19)(15,21)(16,26,36,27)(17,30)(18,33)(20,23)(24,34,37,25)(29,32)(31,35,39,40);(2,6)(3,7)(4,8)(9,11,23,26,15,28,14,27)(10,16,20,22,21,36,19,38)(12,24,30,35,18,37,29,40)(13,31,17,34,33,39,32,25);(9,12)(10,17)(11,24)(13,19)(14,29)(15,18)(16,34)(20,33)(21,32)(22,39)(23,30)(25,36)(26,35)(27,40)(28,37)(31,38);(1,3,8,2,5,6,4,7)(9,13,23,17,15,33,14,32)(10,18,20,29,21,12,19,30)(11,25,27,39,28,34,26,31)(16,35,38,24,36,40,22,37)

Polycyclic presentation:

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

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, 137607);