#109155

order	512
nilpotency class	3
isoclinism	isoclinic class of #94978
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

660 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)(5,6,9,12,10,7,8,11)(13,14)(15,18,21,25)(16,23)(17,19)(20,24)(22,28,27,26);(5,7,10,6)(8,12,9,11)(13,15,16,22,17,21,24,27)(14,18,20,26,19,25,23,28);(1,2,3,4)(5,8,10,9)(6,11,7,12)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ehi,d^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=dh,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16351);