#473995

order	512
nilpotency class	3
isoclinism	isoclinic class of #454664
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

773 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)(9,10,14,16,15,20,13,19)(11,21,22,31,25,33,24,17)(12,26,29,18,30,35,28,36)(23,39,27,37,40,32,38,34);(1,3,4,7)(2,5,6,8)(10,16)(11,22)(13,14)(17,31)(18,26)(19,20)(23,38)(24,25)(27,40)(28,29)(32,39)(35,36);(3,7)(5,8)(9,11,13,24,15,25,14,22)(10,17,16,21,20,31,19,33)(12,27,29,40,30,38,28,23)(18,34,26,32,36,37,35,39);(9,12)(10,18)(11,23)(13,29)(14,28)(15,30)(16,26)(17,32)(19,35)(20,36)(21,37)(22,38)(24,27)(25,40)(31,39)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 381191);