#4456911

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1756 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=i,d^2=f,e^2=h,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, 4364107);