#462663

order	512
nilpotency class	3
isoclinism	isoclinic class of #458129
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

845 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 369859);