#6379027

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342728
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

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, 6286223);