#496063

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=e,d^2=g,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=dhi,g^b=gi,d^c=dh,g^c=gi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265101, #72265116, #72265360, …
maximal normal subgroup of	#72265101, #72265116, #72265360, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403259);