#1335709

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1946 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=f,e^2=hi,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, 1242905);