#864909

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

1762 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=fi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 772105);