#6372373

order	512
nilpotency class	3
isoclinism	isoclinic class of #60367
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	C₂×#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3366 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)(11,19,22,32)(12,16)(13,18)(14,17)(15,21,28,25)(20,33,34,40)(23,29)(24,30)(26,31)(27,35,38,37)(36,39);(9,11,13,25,14,22,26,21)(10,15,18,32,17,28,31,19)(12,20,24,37,23,34,36,35)(16,27,30,40,29,38,39,33);(2,5)(6,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,29)(18,30)(19,33)(21,35)(22,34)(25,37)(26,36)(28,38)(31,39)(32,40);(1,3,4,7)(2,6,5,8)(12,23)(16,29)(20,34)(24,36)(27,38)(30,39)(33,40)(35,37);(10,17)(15,28)(16,29)(18,31)(19,32)(27,38)(30,39)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^b=fi,h^b=hi,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, 6279569);