#381675

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288871);