#844829

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1942 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,23)(19,22,24,21)(25,26,29,32)(27,30,28,31);(2,5)(6,7)(9,11,14,12)(10,13,16,15)(17,19,20,24)(18,21,23,22)(25,27,29,28)(26,30,32,31);(9,11,14,12)(10,15,16,13)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,15,16,13)(17,18)(19,21)(20,23)(22,24)(25,26)(27,30)(28,31)(29,32);(1,2)(3,7)(4,5)(6,8)(9,13,14,15)(10,12,16,11)(17,19)(18,22)(20,24)(21,23)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=fh,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752025);