#4380234

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4486 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,e^2=f,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, 4287430);