#5590551

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336427, #72336449, #72336453
maximal normal subgroup of	#72336427, #72336449, #72336453

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5497747);