#6379205

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4544 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:

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

Polycyclic presentation:

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

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, 6286401);