#404397

order	512
nilpotency class	3
isoclinism	isoclinic class of #394199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

1167 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:

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

Polycyclic presentation:

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

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, 311593);