#3877661

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3661 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)(3,7)(4,5)(6,8)(9,10,13,12)(11,14,15,16)(18,21)(19,23)(25,26,29,32)(27,31,28,30);(2,5)(3,8)(10,12)(14,16)(17,18,20,21)(19,24,23,22)(25,27)(26,30)(28,29)(31,32);(3,8)(6,7)(10,12)(14,16)(19,23)(22,24)(27,28)(30,31);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,24)(20,21)(22,23)(25,26)(27,31)(28,30)(29,32);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,14,15,16)(17,19,20,23)(18,22,21,24)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,e^2=f,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	#72336518, #72336538, #72336598
maximal normal subgroup of	#72336518, #72336538, #72336598

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3784857);