#4112162

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gh,c^2=fhi,d^2=fhi,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, 4019358);