#481474

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=ci,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

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388670);