#898719

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

2420 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)(10,13)(11,15)(18,21)(22,24)(25,26)(27,28)(29,30)(31,32);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,21)(22,24)(26,30)(28,31);(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,30)(29,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,18)(19,22)(20,21)(23,24)(25,28,29,31)(26,27,30,32);(1,4,5,2)(3,8,7,6)(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,22,21,24)(25,27)(26,31)(28,30)(29,32)

Polycyclic presentation:

2^9|b^2=fgi,d^2=f,e^2=fghi,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.

Relationships

maximal subgroup of	#72263208, #72263221, #72263228
maximal normal subgroup of	#72263208, #72263221, #72263228

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 805915);