#878447

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

1676 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fh,d^2=fhi,e^2=g,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, 785643);