#913732

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2393 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=gi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 820928);