#453934

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

589 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)(3,7)(4,6)(5,8)(9,10,15,20)(11,21)(12,13,28,32)(14,18,33,19)(16,25)(17,30,37,31)(22,38)(23,39)(24,36)(26,35)(27,34)(29,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,26)(10,16)(12,27,28,39)(13,29)(14,24,33,25)(17,22,37,40)(18,21)(19,35)(20,36)(23,31)(30,34)(32,38);(2,6)(5,7)(9,12,14,17,15,28,33,37)(10,13,18,30,20,32,19,31)(11,22,25,27,26,40,24,39)(16,34,35,29,36,23,21,38);(3,8)(5,7)(9,13,15,32)(10,17,20,37)(11,23)(12,18,28,19)(14,30,33,31)(16,27)(21,22)(24,29)(25,38)(26,34)(35,40)(36,39)

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=g,d^2=efg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^d=hi

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, 361130);