#4405510

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2259 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,3)(6,9)(10,12)(14,17)(15,19)(21,22)(23,29)(24,26)(25,28)(27,31)(30,36)(32,34)(33,35);(14,17)(18,20)(22,24)(27,29)(28,32)(35,36);(1,2,4,3)(13,14,16,17)(15,18,19,20)(21,23)(22,27)(24,29)(25,33)(26,31)(28,36)(30,34)(32,35);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(21,24,26,22)(23,29,31,27)(25,32,34,28)(30,36,33,35);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,25,26,34)(22,28,24,32)(23,30,31,33)(27,35,29,36)

Polycyclic presentation:

2^9|c^2=fg,d^2=gh,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, 4312706);