#614924

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2224 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=fg,d^2=h,e^2=f,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, 522120);