#6396169

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
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	#60749	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1922 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)(9,10,14,19,15,20,13,17)(11,16,23,30,24,31,22,28)(12,18,27,33,25,34,26,32)(21,29,37,39,35,40,36,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,23,15,24,13,22)(10,16,19,30,20,31,17,28)(12,21,27,37,25,35,26,36)(18,29,33,39,34,40,32,38);(10,17,20,19)(12,25)(16,28,31,30)(18,33,34,32)(21,35)(26,27)(29,39,40,38)(36,37);(3,8)(5,7)(9,12)(10,18)(11,21)(13,26)(14,27)(15,25)(16,29)(17,32)(19,33)(20,34)(22,36)(23,37)(24,35)(28,38)(30,39)(31,40);(2,6)(5,7)(12,25)(18,34)(21,35)(26,27)(29,40)(32,33)(36,37)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6303365);