#6595771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#3515	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6502967);