#156605

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egh,c^2=efg,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 63801);