#4386827

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2066 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)(7,11)(10,12)(15,19)(18,20)(21,22,27,29)(23,30,24,25)(26,28,34,36)(31,35,32,33);(6,9)(10,12)(14,17)(18,20)(21,23,27,24)(22,25,29,30)(26,31,34,32)(28,33,36,35);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,25,29,30)(26,31,34,32)(28,35,36,33);(1,3,4,2)(13,15)(14,18)(16,19)(17,20)(21,25)(22,23)(24,29)(26,33)(27,30)(28,31)(32,36)(34,35);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,26)(22,28)(23,31)(24,32)(25,33)(27,34)(29,36)(30,35)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fhi,d^2=g,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, 4294023);