#531629

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
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

4218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef

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, 438825);