#473994

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

756 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381190);