#373992

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

774 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)(3,5)(4,6)(7,8)(9,10,16,20,15,21,14,34)(11,22,27,37,26,38,25,17)(12,28,31,36,30,18,24,39)(13,32,23,40,29,35,33,19);(1,3,4,7)(2,5,6,8)(9,11,16,27,15,26,14,25)(10,17,34,38,21,37,20,22)(12,29,31,33,30,13,24,23)(18,19,36,35,28,40,39,32);(2,6)(5,8)(9,12,16,31,15,30,14,24)(10,18,20,39,21,28,34,36)(11,23,27,29,26,33,25,13)(17,35,22,19,37,32,38,40);(9,13,14,33,15,29,16,23)(10,19,20,32,21,40,34,35)(11,24,25,30,26,31,27,12)(17,36,22,18,37,39,38,28)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281188);