#364702

order	512
nilpotency class	3
isoclinism	isoclinic class of #196161
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1504 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)(3,6)(4,5)(7,8)(9,10,14,20,15,19,13,18)(11,16,23,32,24,33,22,31)(12,25,27,17,29,34,28,35)(21,36,37,40,26,39,38,30);(2,5)(6,8)(9,11,14,23,15,24,13,22)(10,16,20,32,19,33,18,31)(12,26,27,38,29,21,28,37)(17,30,34,36,35,40,25,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,27)(14,28)(15,29)(16,30)(18,34)(19,35)(20,25)(22,37)(23,38)(24,26)(31,36)(32,39)(33,40);(2,5)(6,8)(9,13,15,14)(10,18,19,20)(11,22,24,23)(12,27,29,28)(16,31,33,32)(17,34,35,25)(21,37,26,38)(30,36,40,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,h^c=hi

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, 271898);