#109111

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

734 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ci,d^b=dh,f^b=fh

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, 16307);