#108933

order	512
nilpotency class	3
isoclinism	isoclinic class of #95118
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

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=gi,d^b=di,f^b=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16129);