#3943283

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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
prime radical	#16249	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3093 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)(3,7,8,4)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,21)(22,24)(26,27)(28,32);(2,6)(4,7)(9,11,12,16)(10,13,14,15)(17,18,20,21)(19,22,24,23)(26,27)(30,31);(2,6)(4,7)(11,16)(13,15)(18,21)(22,23)(25,26,29,27)(28,30,32,31);(1,3,5,8)(2,4,6,7)(9,10)(11,15)(12,14)(13,16)(17,19,20,24)(18,22,21,23)(25,27,29,26)(28,31,32,30);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=fi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232579, #72232679, #72232691
maximal normal subgroup of	#72232579, #72232679, #72232691

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3850479);