#481566

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

865 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:

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

Polycyclic presentation:

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

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, 388762);