#4751472

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

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

3190 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=g,e^2=fghi,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	#72326589, #72326596, #72326643
maximal normal subgroup of	#72326589, #72326596, #72326643

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4658668);