#615495

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2249 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

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, 522691);