#1269470

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

2574 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)(11,13)(12,15)(17,18,22,23)(19,24,26,21)(20,28,30,25)(27,31,32,29);(5,6,7,8)(9,10,14,16)(11,15,13,12)(18,23)(19,26)(25,28)(27,32);(1,2,4,3)(5,6)(7,8)(9,11,14,13)(10,15,16,12)(17,19)(18,24)(20,27)(21,23)(22,26)(25,29)(28,31)(30,32);(9,12)(10,11)(13,16)(14,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,31,24,29);(1,3,4,2)(9,13,14,11)(10,12,16,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,32,28,27)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=g,d^2=f,e^2=fg,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, 1176666);