#6512885

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3562	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(13,14,19,24)(15,25,29,33)(16,22,30,37)(17,23,18,21)(20,28,35,27)(26,39,40,43)(31,38,32,36)(34,42,44,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,24,35)(16,26,30,40)(17,28,18,27)(21,33,23,25)(22,34,37,44)(31,42,32,41)(36,43,38,39);(6,9)(7,11)(14,21)(17,18)(20,33)(22,36)(23,24)(25,35)(27,28)(31,32)(34,43)(37,38)(39,44)(41,42);(5,7,8,11)(6,10,9,12)(13,16,19,30)(14,22,24,37)(15,26,29,40)(17,31,18,32)(20,34,35,44)(21,36,23,38)(25,39,33,43)(27,41,28,42);(6,9)(10,12)(16,30)(22,37)(26,40)(31,32)(34,44)(36,38)(39,43)(41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6420081);