#4746705

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×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

2987 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)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,24)(21,22)(25,26)(27,32)(28,29)(30,31);(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,20,22,23)(25,27,30,29)(26,28,31,32);(11,16)(13,15)(18,22)(20,23)(26,31)(27,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,22)(21,24)(25,28)(26,27)(29,31)(30,32);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,23)(18,19,22,24)(25,29,30,27)(26,28,31,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,e^2=fhi,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.

Relationships

maximal subgroup of	#72328573, #72328643, #72328659
maximal normal subgroup of	#72328573, #72328643, #72328659
maximal quotient of	#72330213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4653901);