#5784109

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

3097 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)(9,10,12,14)(11,15,16,13)(18,22)(19,23)(25,26)(27,30)(28,29)(31,32);(3,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,20,23,24)(25,27)(26,30)(28,31)(29,32);(2,5)(6,8)(11,16)(13,15)(18,22)(19,23)(25,27,29,32)(26,31,28,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,31)(28,30)(29,32);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,14)(13,16)(17,20)(18,23)(19,22)(21,24)(25,28,29,26)(27,30,32,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,c^2=fi,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	#72232584, #72232596, #72232640
maximal normal subgroup of	#72232584, #72232596, #72232640
maximal quotient of	#72328654

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5691305);