#397874

order	512
nilpotency class	3
isoclinism	isoclinic class of #394382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1007 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ei,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,h^d=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, 305070);