#1388638

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2631 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fghi,d^2=f,e^2=ghi,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, 1295834);