#615891

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

2219 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)(4,8)(10,15)(13,16)(18,23)(19,21)(25,26)(27,29)(28,31)(30,32);(1,2,5,3)(4,6,8,7)(9,10,14,15)(11,16,12,13)(17,18)(19,20)(21,24)(22,23)(26,27)(30,31);(9,11,14,12)(10,16,15,13)(17,19,22,21)(18,20,23,24)(25,27,29,26)(28,31,32,30);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,15,16)(17,20,22,24)(18,19,23,21);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,15)(14,16)(17,21,22,19)(18,24,23,20)(25,28,29,32)(26,30,27,31)

Polycyclic presentation:

2^9|b^2=fg,c^2=fi,d^2=fh,e^2=g,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, 523087);