#363633

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

661 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270829);