#4616416

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3184 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)(4,8)(5,6)(9,10,13,14)(11,15,16,12)(17,18,20,22)(19,23,24,21)(25,26,28,30)(27,29,31,32);(2,6)(3,8)(9,11)(10,12)(13,16)(14,15)(17,18,20,22)(19,21,24,23)(25,27,28,31)(26,29,30,32);(2,6)(4,7)(11,16)(12,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,29)(28,31)(30,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,22)(23,24);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4523612);