#4084645

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1986 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)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,16,12)(18,19)(20,24)(25,26,28,30)(27,31,32,29);(2,4)(3,7)(9,11,13,16)(10,12,14,15)(20,24)(22,23)(25,26,28,30)(27,31,32,29);(2,4)(6,8)(11,16)(12,15)(17,18,21,19)(20,22,24,23)(25,27,28,32)(26,29,30,31);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,18)(20,23,24,22);(1,4,5,2)(3,6,7,8)(9,12,13,15)(10,11,14,16)(17,20,21,24)(18,22,19,23)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=fgh,e^2=gh,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, 3991841);