#412059

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1362 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)(6,7)(9,10,16,22)(11,17,25,35)(12,26,29,40)(13,30,32,38)(14,21,15,20)(18,33,37,28)(19,31,39,23)(24,36,27,34);(5,6)(7,8)(9,11)(10,17)(12,13)(14,27)(15,24)(16,25)(18,19)(20,36)(21,34)(22,35)(23,28)(26,30)(29,32)(31,33)(37,39)(38,40);(2,4)(5,7,8,6)(9,12)(10,18)(11,23)(13,27)(14,33)(15,28)(16,29)(17,30)(19,36)(20,40)(21,26)(22,37)(24,32)(25,31)(34,39)(35,38);(9,13,14,31,16,32,15,23)(10,19,20,38,22,39,21,30)(11,12,27,33,25,29,24,28)(17,18,36,40,35,37,34,26)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,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, 319255);