#4130952

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2253 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gh,c^2=fhi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303943, #72304111, #72304113
maximal normal subgroup of	#72303943, #72304111, #72304113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4038148);