#106275

order	512
nilpotency class	3
isoclinism	isoclinic class of #94978
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×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	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6,8,12,10,14,9,13)(7,15,16,20,18,19,17,11)(21,22,25,30,26,29,24,28)(23,27,31,34,32,35,33,36);(5,7,9,17,10,18,8,16)(6,11,13,19,14,20,12,15)(21,23)(22,27,29,35)(24,31)(25,33)(26,32)(28,34,30,36);(2,4)(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,15,20,19)(21,24,26,25)(22,28,29,30)(23,31,32,33)(27,34,35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,f^2=h,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, 13471);