#4414504

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2688 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)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,29)(30,31);(9,11,14,12)(10,15,13,16)(18,23)(19,25)(26,31)(27,29);(5,6,7,8)(9,12,14,11)(10,15,13,16)(17,19,22,25)(18,24,23,20)(21,29,30,27)(26,28,31,32);(1,2)(3,4)(17,20,22,24)(18,25,23,19)(21,28,30,32)(26,29,31,27);(9,13,14,10)(11,16,12,15)(17,21,22,30)(18,26,23,31)(19,27,25,29)(20,28,24,32)

Polycyclic presentation:

2^9|b^2=fh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,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, 4321700);