#2305996

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×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2282 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,22,20)(19,24,21,23)(25,26,31,32)(27,28,30,29);(2,3)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,21)(18,23,20,24)(25,27,31,30)(26,29,32,28);(4,8)(6,7)(11,16)(13,14)(17,18,22,20)(19,23,21,24)(25,28,31,29)(26,27,32,30);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,20,22,18)(19,23,21,24)(25,29,31,28)(26,27,32,30);(1,4,5,8)(2,6,3,7)(9,11)(10,14)(12,16)(13,15)(17,21,22,19)(18,23,20,24)(25,30,31,27)(26,28,32,29)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fghi,c^2=fghi,d^2=fgh,e^2=fgh,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, 2213192);