#4625810

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2914 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)(3,7)(9,10,14,15)(11,16,13,12)(17,18,22,21)(19,20,24,23)(25,26,28,29)(27,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,12,15,16)(17,19)(18,23)(20,21)(22,24)(26,29)(30,31);(3,7)(6,8)(10,15)(11,13)(18,21)(19,24)(25,27,28,32)(26,30,29,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,12,15,16)(17,21,22,18)(19,20,24,23)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=gh,e^2=fgh,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	#72336295, #72336402, #72336445
maximal normal subgroup of	#72336295, #72336402, #72336445

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4533006);