#369651

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

909 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)(10,15,18,26)(12,23,25,24)(16,19,30,34)(20,35,37,36);(9,10,13,26,14,18,27,15)(11,19,21,16,22,34,38,30)(12,17,24,31,25,32,23,28)(20,33,36,29,37,39,35,40);(2,5)(6,8)(9,11,13,21,14,22,27,38)(10,16,15,19,18,30,26,34)(12,20,24,36,25,37,23,35)(17,29,28,33,32,40,31,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,24)(14,25)(15,28)(16,29)(18,32)(19,33)(21,36)(22,37)(23,27)(26,31)(30,40)(34,39)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=g,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^d=ei,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, 276847);