#4342205

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2699	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3132 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)(6,7)(11,15)(14,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31);(10,13)(14,16)(18,21)(19,25)(24,28)(26,29);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,25)(18,23,21,27)(20,29,30,26)(24,32,28,31);(1,2,3,4)(5,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32);(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,27,25,23)(20,28,30,24)(26,32,29,31)

Polycyclic presentation:

2^9|c^2=fg,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4249401);