#865719

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2009 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgh,c^2=fgi,e^2=fg,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, 772915);