#517335

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
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

2454 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)(5,6,7,8)(10,13)(14,16)(17,18,22,23)(19,26,25,21)(20,24,30,31)(27,32,28,29);(1,2,4,3)(10,13)(14,16)(18,23)(19,25)(24,31)(27,28);(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,22,25)(18,21,23,26)(20,28,30,27)(24,32,31,29);(17,20,22,30)(18,24,23,31)(19,27,25,28)(21,29,26,32);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,23)(20,29)(22,26)(24,28)(27,31)(30,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=h,d^2=f,e^2=gi,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 424531);