#6629177

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318387, #72318395, #72318436, …
maximal normal subgroup of	#72318387, #72318395, #72318436, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536373);