#474873

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

536 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)(3,7)(4,5)(6,8)(9,10,15,20)(11,21,25,32)(12,26)(13,19,14,18)(16,24,33,23)(17,28)(22,37)(27,31)(29,34)(30,35)(36,38)(39,40);(2,5)(6,7)(9,11,13,23,15,25,14,24)(10,16,18,21,20,33,19,32)(12,27,28,39,30,22,29,38)(17,31,26,36,34,37,35,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,30)(10,17)(11,22,25,27)(13,28,14,29)(16,31)(18,26)(19,35)(20,34)(21,36)(23,38,24,39)(32,40)(33,37);(3,8)(6,7)(10,18,20,19)(12,28,30,29)(16,21,33,32)(17,34)(22,38,27,39)(26,35)(31,37)(36,40)

Polycyclic presentation:

2^9|a^2=efg,b^2=efgh,c^2=g,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,h^c=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, 382069);