#550055

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

2226 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fi,d^2=fg,e^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 457251);