#390957

order	512
nilpotency class	3
isoclinism	isoclinic class of #232169
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
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

809 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)(3,5)(4,6)(7,8)(9,10,14,19)(11,20)(12,17)(13,15,27,18)(16,22)(21,36,39,35)(23,32)(24,29)(25,34)(26,33)(28,30)(31,38,40,37);(1,3,5,2)(4,7,8,6)(10,15)(12,25)(13,27)(16,20)(17,33)(18,19)(21,37)(22,23)(26,28)(29,32)(35,36)(38,39);(1,4,5,8)(2,6,3,7)(9,11,13,22,14,24,27,23)(10,16,15,29,19,32,18,20)(12,21,28,38,26,39,25,37)(17,31,30,36,33,40,34,35);(2,3)(6,7)(9,12)(10,17)(11,21)(13,28)(14,26)(15,30)(16,31)(18,34)(19,33)(20,35)(22,38)(23,37)(24,39)(25,27)(29,36)(32,40)

Polycyclic presentation:

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

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, 298153);