#553749

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 460945);