#295376

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=i,c^2=f,d^2=ehi,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202572);