#4929343

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
automorphism	Aut(G) order 2097152
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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2878 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222747
maximal normal subgroup of	#72222747
maximal quotient of	#72222692

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4836539);