#540394

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2688 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)(9,10,14,16)(11,12,13,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32);(5,6,7,8)(9,11,14,13)(10,12,16,15)(18,22)(20,29)(23,25)(26,32);(5,6)(7,8)(9,12,14,15)(10,11,16,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(9,10)(11,12)(13,15)(14,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31);(1,3,4,2)(9,13,14,11)(10,15,16,12)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=g,d^2=f,e^2=fi,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 447590);