#4448532

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1928 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=fg,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4355728);