#899377

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2178 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=g,e^2=fghi,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, 806573);