#4389773

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2939 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)(10,14)(12,15)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32);(9,10)(11,12)(13,14)(15,16)(18,23)(20,29)(24,30)(26,32);(5,6,8,7)(10,14)(12,15)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32);(1,2,3,4)(5,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24);(9,12,13,15)(10,11,14,16)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)

Polycyclic presentation:

2^9|a^2=fh,c^2=g,d^2=i,e^2=h,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, 4296969);