#445125

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

957 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)(3,6)(4,5)(7,8)(9,10,13,18,14,17,27,33)(11,19,21,34,22,15,26,29)(12,16,25,32,24,31,39,40)(20,35,23,37,36,28,38,30);(9,11)(10,15)(12,23)(13,26)(14,22)(16,30)(17,19)(18,34)(20,25)(21,27)(24,38)(28,32)(29,33)(31,37)(35,40)(36,39);(2,5)(6,8)(9,12,13,25,14,24,27,39)(10,16,18,32,17,31,33,40)(11,20,21,23,22,36,26,38)(15,28,29,30,19,35,34,37);(1,3,4,7)(2,6,5,8)(12,24)(15,19)(16,31)(20,36)(23,38)(25,39)(29,34)(32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 352321);