#1293309

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1654 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)(3,7)(4,5)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,20)(19,24,21,23)(25,26,30,32)(27,28,31,29);(1,3)(2,6)(4,7)(5,8)(9,11,15,14)(10,13,16,12)(17,18,22,20)(19,23,21,24)(25,27,30,31)(26,28,32,29);(9,12,15,13)(10,11,16,14)(17,19,22,21)(18,23,20,24)(25,26)(27,28)(29,31)(30,32);(1,4,5,2)(3,7,8,6)(9,13,15,12)(10,11,16,14)(17,20,22,18)(19,24,21,23)(25,28,30,29)(26,27,32,31);(1,3)(2,7)(4,6)(5,8)(9,14,15,11)(10,13,16,12)(17,21,22,19)(18,24,20,23)(25,29,30,28)(26,31,32,27)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fghi,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1200505);