#528340

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	C₂×#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2974 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 435536);