#4615074

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

3015 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,e^2=g,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	#72327634, #72327659, #72327698
maximal normal subgroup of	#72327634, #72327659, #72327698
maximal quotient of	#72328976

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4522270);