#363641

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=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, 270837);