#6421821

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2714	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6329017);