#106285

order	512
nilpotency class	3
isoclinism	isoclinic class of #94978
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

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	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1060 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)(5,6,9,12)(7,10,11,8)(13,14,17,22,18,21,16,20)(15,19,23,26,24,27,25,28);(5,7,10,6,9,11,8,12)(13,15)(14,19,21,27)(16,23)(17,25)(18,24)(20,26,22,28);(2,4)(5,8,9,10)(6,7,12,11)(13,16,18,17)(14,20,21,22)(15,23,24,25)(19,26,27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,f^2=i,g^2=hi,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, 13481);