#513896

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂²	C₂³×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2582 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)(5,6,8,7)(10,13)(11,15)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31);(5,6,8,7)(10,13)(14,16)(18,22)(19,26)(24,28)(27,32);(1,2,4,3)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,2,4,3)(5,6,8,7)(9,11,12,15)(10,14,13,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31);(1,3,4,2)(5,7,8,6)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=fgh,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421092);