#5228463

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 4194304
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
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2271 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,23)(19,24,22,20)(25,26,29,31)(27,28,32,30);(1,3,5,8)(2,6,7,4)(9,10,13,14)(11,15,16,12)(17,19,21,22)(18,20,23,24)(25,27)(26,30)(28,31)(29,32);(3,8)(4,6)(11,16)(12,15)(17,20,21,24)(18,22,23,19)(25,28,29,30)(26,27,31,32);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26)(27,28)(29,31)(30,32);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262805, #72262806, #72262831
maximal normal subgroup of	#72262805, #72262806, #72262831

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5135659);