#6392722

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂³×C₄⋊C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1934 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)(3,7)(4,6)(5,8)(9,10,14,18,15,19,13,27)(11,16,23,31,21,30,22,37)(12,24,25,17,26,32,28,33)(20,34,36,29,35,40,38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,29)(18,31)(19,30)(24,34)(25,36)(26,35)(27,37)(28,38)(32,40)(33,39);(9,12,14,25,15,26,13,28)(10,17,18,32,19,33,27,24)(11,20,23,36,21,35,22,38)(16,29,31,40,30,39,37,34);(3,8)(5,7)(11,21)(16,30)(20,35)(22,23)(29,39)(31,37)(34,40)(36,38);(2,6)(5,7)(11,21)(16,30)(20,35)(22,23)(29,39)(31,37)(34,40)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315518, #72315619, #72315625
maximal normal subgroup of	#72315518, #72315619, #72315625

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6299918);