#896060

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

2364 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=fh,e^2=i,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, 803256);