#462100

order	512
nilpotency class	3
isoclinism	isoclinic class of #454676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369296);