#4456264

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2219 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4363460);