#1846137

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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

1839 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fi,d^2=fghi,e^2=hi,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, 1753333);