#411690

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1844 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318886);