#4435228

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2661	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3175 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)(13,16)(17,18,22,23)(19,21,25,31)(20,27,29,24)(26,28,32,30);(9,10,12,14)(11,13,15,16)(18,23)(19,25)(24,27)(26,32);(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,21)(20,26)(22,25)(23,31)(24,30)(27,28)(29,32);(1,2,3,4)(5,7,8,6)(17,20,22,29)(18,24,23,27)(19,26,25,32)(21,30,31,28);(9,11,12,15)(10,13,14,16)(17,21)(18,19)(20,28)(22,31)(23,25)(24,32)(26,27)(29,30)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=g,d^2=i,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, 4342424);