#146632

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×Q₈	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,4)(9,10,15,21,16,22,14,20)(11,23,26,37,27,39,25,33)(12,19,30,18,13,31,29,38)(17,32,35,28,36,40,34,24);(5,6,7,8)(9,11)(10,17,22,36)(12,28)(13,24)(14,26)(15,25)(16,27)(18,37,38,33)(19,23,31,39)(20,34,21,35)(29,40)(30,32);(2,4)(9,12,16,13)(10,18,22,38)(11,24,27,28)(14,30,15,29)(17,33,36,37)(19,20,31,21)(23,35,39,34)(25,32,26,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fhi,d^c=dh,e^c=ei,f^c=fi,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, 53828);