#539139

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=i,e^2=g,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, 446335);