#105457

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=h,g^2=i,b^a=bd,d^a=dh,f^a=fhi,c^b=chi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12653);