#4329052

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2781 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)(10,11)(12,16)(17,18,20,21)(19,23,24,22)(25,26,29,28)(27,31,32,30);(2,6)(4,7)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,21)(22,24)(26,28)(30,31);(2,6)(3,8)(10,11)(14,15)(18,21)(22,23)(25,27,29,32)(26,30,28,31);(1,3)(2,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32);(1,4,5,7)(2,3,6,8)(9,12)(10,14)(11,15)(13,16)(17,19,20,24)(18,22,21,23)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=hi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336429, #72336449, #72336456
maximal normal subgroup of	#72336429, #72336449, #72336456

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4236248);