#428021

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1305 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:

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

Polycyclic presentation:

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

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, 335217);