#367714

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
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	#2358
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

951 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)(9,10)(11,22,25,36)(12,17)(13,19)(14,20)(15,21)(16,24,18,29)(23,38,40,32)(26,31,39,33)(27,34)(28,35)(30,37);(9,11)(10,16)(12,26,28,39)(13,29)(14,24)(15,25)(17,32,35,38)(18,21)(19,36)(20,22)(23,27,40,30)(31,34,33,37);(2,5)(6,8)(9,12)(10,17)(11,23)(13,27)(14,30)(15,28)(16,31)(18,33)(19,34)(20,37)(21,35)(22,38)(24,39)(25,40)(26,29)(32,36);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,23)(13,29)(14,24)(15,25)(16,21)(17,33)(19,22)(20,36)(26,27)(28,40)(30,39)(31,35)(32,37)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274910);