#463538

order	512
nilpotency class	3
isoclinism	isoclinic class of #454986
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

915 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:

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

Polycyclic presentation:

2^9|a^2=efg,c^2=ef,d^2=fi,g^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,g^b=gi,d^c=de,g^c=gi,h^c=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, 370734);