#1440241

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2297 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)(3,7)(9,10,14,15)(11,12,16,13)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,30)(29,32);(1,2)(3,8)(4,5)(6,7)(9,11,14,16)(10,13,15,12)(18,21)(22,23)(25,26,28,30)(27,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,11,15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,8)(4,5)(6,7)(9,13,14,12)(10,11,15,16)(17,19,20,24)(18,22,21,23)(25,27,28,32)(26,29,30,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,21)(22,24)(25,26)(27,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fgh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

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, 1347437);