#453882

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	#2495
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

544 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fh,c^2=g,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 361078);