#423648

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1570 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)(7,8)(9,10,16,22)(11,23,27,38)(12,28,30,19)(13,31,29,18)(14,21,15,20)(17,33,36,26)(24,35,25,34)(32,37,40,39);(1,2)(3,4)(9,11)(10,17)(12,24)(13,32)(14,33)(15,26)(16,27)(18,34)(19,37)(20,38)(21,23)(22,36)(25,30)(28,39)(29,40)(31,35);(2,4)(9,12,14,13,16,30,15,29)(10,18,20,19,22,31,21,28)(11,24,33,32,27,25,26,40)(17,34,38,37,36,35,23,39);(6,8)(9,13,14,30,16,29,15,12)(10,19,20,31,22,28,21,18)(11,25,26,32,27,24,33,40)(17,35,23,37,36,34,38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330844);