#1506808

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2644 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)(11,13)(12,15)(14,16)(17,18)(19,24)(20,27)(21,23)(22,29)(25,30)(26,32)(28,31);(6,8)(10,11)(12,16)(17,19,23,20)(18,24,21,27)(22,26,30,28)(25,31,29,32);(5,6)(7,8)(10,11)(14,15)(17,20,23,19)(18,24,21,27)(22,28,30,26)(25,31,29,32);(1,2)(3,4)(9,11,13,10)(12,15,16,14)(17,21,23,18)(19,24,20,27)(22,29,30,25)(26,31,28,32);(9,12,13,16)(10,14,11,15)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)

Polycyclic presentation:

2^9|b^2=fg,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1414004);