#474700

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=gi,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, 381896);