#1293327

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

1815 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)(6,7)(9,10,12,15)(11,14,16,13)(17,18,22,20)(19,24,21,23)(25,26)(27,32)(28,30)(29,31);(1,2,5,3)(4,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,18,22,20)(19,23,21,24)(25,27)(26,28)(29,30)(31,32);(9,11)(10,13)(12,16)(14,15)(17,19,22,21)(18,23,20,24)(25,26)(27,32)(28,30)(29,31);(1,3,5,2)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,20,22,18)(19,24,21,23)(25,28,29,32)(26,30,31,27);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,15)(14,16)(17,21,22,19)(18,24,20,23)(25,26)(27,28)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=fghi,d^2=fg,e^2=fgh,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, 1200523);