#453930

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

587 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:

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

Polycyclic presentation:

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

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, 361126);