#397873

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1039 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)(9,10,16,29,14,13,15,19)(11,20,24,32,22,17,23,35)(12,25,30,18,27,33,28,26)(21,36,39,34,37,40,38,31);(1,3,5,4)(2,6,8,7)(9,11,16,24,14,22,15,23)(10,17,29,35,13,20,19,32)(12,21,30,39,27,37,28,38)(18,31,33,36,26,34,25,40);(1,3,5,4)(2,7,8,6)(9,12,14,27)(10,18,13,26)(11,21,22,37)(15,30,16,28)(17,31,20,34)(19,33,29,25)(23,39,24,38)(32,36,35,40);(1,4,5,3)(2,6,8,7)(9,13)(10,14)(11,20)(12,26)(15,29)(16,19)(17,22)(18,27)(21,34)(23,35)(24,32)(25,30)(28,33)(31,37)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,d^2=ei,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, 305069);