#4379772

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×#2635	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3426 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)(14,16)(18,19)(20,27)(24,26)(28,32);(10,13)(14,16)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31);(1,2)(3,4)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^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, 4286968);