#6384172

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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	#61124	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1985 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:

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

Polycyclic presentation:

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

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, 6291368);