#422495

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1293 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329691);