#4422950

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

4140 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)(6,8)(11,15)(13,16)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32);(9,10,12,14)(11,13,15,16)(18,22)(23,25)(24,28)(31,32);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32);(1,2)(3,4)(5,6)(7,8)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31);(9,11,12,15)(10,13,14,16)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fh,e^2=fh,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, 4330146);