#556962

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=g,d^2=fi,e^2=i,b^a=bf,c^a=cg,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, 464158);