#536855

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3060 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=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, 444051);