#4311885

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#16500	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=fg,d^2=fg,e^2=f,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4219081);