#1311543

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2328 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)(9,10,14,11)(12,16,13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,30)(31,32);(5,6,7,8)(9,11,14,10)(12,16,13,15)(18,23)(19,27)(25,28)(26,32);(1,2,4,3)(5,6)(7,8)(9,12,14,13)(10,15,11,16)(17,19,22,27)(18,24,23,29)(20,26,30,32)(21,28,31,25);(9,12,14,13)(10,15,11,16)(17,20)(18,25)(19,26)(21,24)(22,30)(23,28)(27,32)(29,31);(1,3,4,2)(9,13,14,12)(10,16,11,15)(17,21)(18,26)(19,28)(20,29)(22,31)(23,32)(24,30)(25,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,d^2=i,e^2=g,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, 1218739);