#516377

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×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

3086 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)(7,11)(8,9)(10,12)(13,14)(15,20)(16,17)(18,19)(22,24)(26,27);(6,9)(10,11)(14,19)(15,16)(21,22,25,24)(23,27,28,26);(1,2,3,4)(13,15,18,16)(14,20,19,17)(21,22,25,24)(23,26,28,27);(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,17,19,20)(21,23,25,28)(22,26,24,27);(5,7)(6,10)(8,12)(9,11)(13,17)(14,15)(16,19)(18,20)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=gh,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=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, 423573);