#4409983

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2005 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gh,c^2=fgh,d^2=fgh,e^2=fgi,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, 4317179);