#453868

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

609 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361064);