#453925

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
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

575 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)(11,21,25,32)(12,17)(13,19)(14,18)(15,20)(16,24,31,23)(22,37,39,33)(26,40,38,30)(27,35)(28,34)(29,36);(1,3,5,4)(2,6,7,8)(9,11,15,25)(10,16,20,31)(12,26,29,38)(13,23,14,24)(17,33,36,37)(18,21,19,32)(22,27,39,28)(30,34,40,35);(2,7)(6,8)(9,12,13,27,15,29,14,28)(10,17,19,35,20,36,18,34)(11,22,24,38,25,39,23,26)(16,30,21,33,31,40,32,37);(1,4,5,3)(2,8,7,6)(10,18,20,19)(11,23,25,24)(17,34,36,35)(22,26,39,38)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361121);