#474699

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

566 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=g,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^d=gi,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, 381895);