#4423129

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2180 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=fh,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4330325);