#517571

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=i,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 424767);