#3333868

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,10)(8,12)(9,11)(13,14,16,18)(15,19,20,17)(21,22,26,27)(23,24,29,28)(25,33,36,30)(31,35,34,32);(5,7,9,12)(6,10,11,8)(13,14,16,18)(15,17,20,19)(22,27)(23,29)(30,33)(31,34);(5,7)(6,8)(9,12)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23,26,29)(22,28,27,24)(25,34,36,31)(30,32,33,35);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,24,26,28)(22,29,27,23)(25,35,36,32)(30,31,33,34);(5,8)(6,7)(9,10)(11,12)(13,15)(14,17)(16,20)(18,19)(21,25,26,36)(22,30,27,33)(23,31,29,34)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304103, #72304122, #72304129
maximal normal subgroup of	#72304103, #72304122, #72304129

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3241064);