#517635

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2902 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=i,e^2=g,b^a=bf,c^a=cg,c^b=ch,e^d=ef

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, 424831);