#4420355

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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

4086 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,e^2=h,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, 4327551);