#1348038

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2756 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fh,e^2=fi,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, 1255234);