#579610

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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

2824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,3)(10,12)(14,16)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28);(1,2,4,3)(6,8)(11,15)(14,16)(18,23)(19,25)(24,27)(26,31);(5,6)(7,8)(9,10,13,12)(11,14,15,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,28)(27,32)(29,31);(1,3,4,2)(9,11)(10,14)(12,16)(13,15)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32);(9,12,13,10)(11,16,15,14)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=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, 486806);