#4387660

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2508 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)(3,4)(5,7)(6,8)(10,13)(14,15)(17,18,22,23)(19,21,20,24)(25,26,29,31)(27,28,32,30);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,13)(14,16)(18,23)(19,20)(25,27,29,32)(26,30,31,28);(2,7)(4,6)(10,13)(14,15)(17,19,22,20)(18,21,23,24)(25,28,29,30)(26,27,31,32);(1,3)(2,4)(5,8)(6,7)(9,11,12,16)(10,14,13,15)(17,20,22,19)(18,21,23,24)(25,26)(27,28)(29,31)(30,32);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,19)(20,23)(22,24)(25,28)(26,27)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4294856);