#2040408

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2143 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1947604);