#2073541

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1765 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)(3,8)(4,7)(5,6)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,20,23,22)(26,29)(30,31);(1,3,6,4)(2,7,5,8)(11,16)(13,15)(19,23)(20,22)(25,26,28,29)(27,31,32,30);(9,11)(10,13)(12,16)(14,15)(17,19,21,23)(18,22,24,20)(25,27,28,32)(26,30,29,31);(1,4,6,3)(2,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,22)(18,23,24,19);(1,5,6,2)(3,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=hi,d^2=fhi,e^2=fgh,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, 1980737);