#4746688

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3011 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4653884);