#285125

order	512
nilpotency class	3
isoclinism	isoclinic class of #285115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18766	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

786 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)(3,5)(4,7)(6,8)(9,10,14,19,15,20,13,18)(11,16,23,32,24,33,22,31)(12,25,27,35,28,36,29,17)(21,37,38,34,26,40,39,30);(2,5)(3,8)(6,7)(9,11,14,23,15,24,13,22)(10,16,19,32,20,33,18,31)(12,26,27,39,28,21,29,38)(17,34,25,40,35,30,36,37);(9,12,15,28)(10,17,20,35)(11,21,24,26)(13,27,14,29)(16,30,33,34)(18,25,19,36)(22,38,23,39)(31,37,32,40);(2,5)(3,8)(6,7)(9,13,15,14)(10,18,20,19)(11,22,24,23)(12,27,28,29)(16,31,33,32)(17,25,35,36)(21,38,26,39)(30,37,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192321);