#1259646

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

2596 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fg,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, 1166842);