#108422

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,e^a=eh,e^c=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15618);