#6375675

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
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	C₂²×C₄×D₄
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1973 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)(9,10,14,19,15,12,24,18)(11,16,23,30,22,20,33,29)(13,17,27,32,26,25,37,31)(21,28,35,39,36,34,40,38);(2,5)(3,7)(9,11,14,23,15,22,24,33)(10,16,19,30,12,20,18,29)(13,21,27,35,26,36,37,40)(17,28,32,39,25,34,31,38);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,20)(13,25)(14,18)(16,22)(17,26)(19,24)(21,34)(23,29)(27,31)(28,36)(30,33)(32,37)(35,38)(39,40);(2,5)(6,8)(9,13)(10,17)(11,21)(12,25)(14,27)(15,26)(16,28)(18,31)(19,32)(20,34)(22,36)(23,35)(24,37)(29,38)(30,39)(33,40);(11,22)(13,26)(16,20)(17,25)(23,33)(27,37)(29,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6282871);