#5894375

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2961 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)(3,6)(4,5)(7,8)(10,11)(14,15)(17,18)(19,23)(20,21)(22,24)(26,27)(28,32);(3,7)(6,8)(9,10,13,11)(12,14,16,15)(17,19,22,20)(18,23,24,21)(25,26,29,27)(28,31,32,30);(2,5)(6,8)(10,11)(14,15)(18,24)(19,20)(25,26,29,27)(28,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,14,16,15)(17,20,22,19)(18,23,24,21)(25,27,29,26)(28,30,32,31);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,21,22,23)(18,19,24,20)(25,28,29,32)(26,30,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300883, #72300907, #72300914
maximal normal subgroup of	#72300883, #72300907, #72300914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5801571);