#2046670

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1494 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ghi,c^2=fg,d^2=fh,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, 1953866);