#550078

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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

2398 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)(9,10,12,14)(11,13,16,15)(17,18)(19,21)(20,24)(22,23)(25,30)(26,28)(27,32)(29,31);(1,2,4,3)(6,8)(9,10,12,14)(11,15,16,13)(18,23)(19,25)(24,31)(26,27);(1,2,4,3)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28);(9,11)(10,13)(12,16)(14,15)(17,20,22,29)(18,24,23,31)(19,26,25,27)(21,28,30,32);(1,3,4,2)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=f,e^2=gi,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, 457274);