#105465

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×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

748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=h,f^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, 12661);