#562752

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2584 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)(9,10,12,13)(11,15,16,14)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32);(6,8)(10,13)(11,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32);(5,6)(7,8)(10,13)(14,15)(17,20,22,19)(18,23,25,26)(21,27,30,28)(24,32,29,31);(1,3,4,2)(9,10)(11,14)(12,13)(15,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32);(9,11,12,16)(10,14,13,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)

Polycyclic presentation:

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

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, 469948);