#615753

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=fi,d^2=f,e^2=fg,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, 522949);