#4068812

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2013 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)(3,8)(4,7)(5,6)(9,10,13,12)(11,15,16,14)(17,18,21,22)(19,20,24,23)(25,26,29,32)(27,28,31,30);(1,3,6,7)(2,4,5,8)(11,16)(14,15)(18,22)(20,23)(25,27,29,31)(26,30,32,28);(2,5)(3,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,23,22,20)(25,26,29,32)(27,30,31,28);(1,4,6,8)(2,7,5,3)(9,12,13,10)(11,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,32)(29,31);(1,5,6,2)(3,4,7,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,23)(18,19,22,24)(25,28,29,30)(26,31,32,27)

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=i,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3976008);