#363640

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=g,f^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, 270836);