#6463033

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60639	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370229);