#531373

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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

2382 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=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, 438569);