#1435448

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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

1944 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)(3,7,4,8)(10,13)(11,15)(17,18)(19,22)(20,21)(23,24)(27,31)(30,32);(2,5)(7,8)(18,20)(22,24)(25,26,29,28)(27,30,31,32);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,23,24)(25,27)(26,30)(28,32)(29,31);(1,4,6,3)(2,7,5,8)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,24)(21,23)(25,28,29,26)(27,32,31,30);(1,5,6,2)(3,8,4,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,22,23,24)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

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, 1342644);