#5782870

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3235 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fhi,c^2=fi,d^2=fg,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	#72336822, #72336842, #72336849
maximal normal subgroup of	#72336822, #72336842, #72336849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5690066);