#109217

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

790 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di

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, 16413);