#6373531

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^b=ei,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280727);