#868718

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2551 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=hi,d^2=gh,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, 775914);