#493648

order	512
nilpotency class	4
isoclinism	isoclinic class of #493638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20609
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#63041	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232878, #72233665, #72234103
maximal normal subgroup of	#72232878, #72233665, #72234103
maximal quotient of	#72232729, #72233285, #72238426, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400844);