#463896

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
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

809 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=ef,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi,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, 371092);