#146659

order	512
nilpotency class	4
isoclinism	isoclinic class of #12778
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×Q₈	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1009 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53855);