#4420534

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#548	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4672 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281221, #72281298, #72281402
maximal normal subgroup of	#72281221, #72281298, #72281402

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4327730);