#2527353

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2164 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,12,15)(17,18,20,22)(19,23,24,21)(25,26,31,32)(27,29,30,28);(1,3)(2,4)(5,8)(6,7)(9,11,13,12)(10,14,16,15)(17,18,20,22)(19,21,24,23)(25,27,31,30)(26,28,32,29);(2,6)(4,7)(17,18)(19,21)(20,22)(23,24)(25,28,31,29)(26,27,32,30);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,29,31,28)(26,27,32,30);(1,4)(2,3)(5,7)(6,8)(9,12,13,11)(10,15,16,14)(17,19,20,24)(18,21,22,23)(25,30,31,27)(26,28,32,29)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=fgh,d^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 2434549);