#539755

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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

3660 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=gi,e^2=h,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, 446951);