#1605416

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2442 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1512612);