#6372219

order	512
nilpotency class	3
isoclinism	isoclinic class of #60454
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×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	C₂³×C₂²⋊C₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3390 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)(4,5)(6,8)(9,10,14,18,15,19,13,28)(11,20,24,30,22,34,23,16)(12,17,27,32,25,31,26,38)(21,33,37,39,35,40,36,29);(2,5)(6,7)(9,11)(10,16,19,30)(12,21)(13,23)(14,24)(15,22)(17,29,31,39)(18,20,28,34)(25,35)(26,36)(27,37)(32,33,38,40);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,29)(18,32)(19,31)(20,33)(22,35)(23,36)(24,37)(28,38)(30,39)(34,40);(12,25)(17,31)(21,35)(26,27)(29,39)(32,38)(33,40)(36,37);(11,22)(16,30)(20,34)(21,35)(23,24)(29,39)(33,40)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279415);