#6396165

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
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	#60741	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 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)(9,10,14,18,15,19,13,29)(11,16,23,32,21,31,22,38)(12,24,28,34,26,40,27,17)(20,35,37,33,25,39,36,30);(1,3)(2,5)(4,8)(6,7)(9,11,14,23,15,21,13,22)(10,16,18,32,19,31,29,38)(12,25,28,36,26,20,27,37)(17,33,24,39,34,30,40,35);(9,12)(10,17,19,34)(11,20)(13,27)(14,28)(15,26)(16,30,31,33)(18,24,29,40)(21,25)(22,36)(23,37)(32,35,38,39);(3,8)(5,7)(12,26)(17,34)(20,25)(24,40)(27,28)(30,33)(35,39)(36,37);(2,6)(5,7)(11,21)(16,31)(20,25)(22,23)(30,33)(32,38)(35,39)(36,37)

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,c^b=ci,d^b=df,e^b=ei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6303361);