#6377777

order	512
nilpotency class	3
isoclinism	isoclinic class of #60750
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	#60790	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313983, #72315637, #72315641
maximal normal subgroup of	#72313983, #72315637, #72315641

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6284973);