#5035106

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 8388608
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

2288 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)(6,7)(10,11)(13,14)(15,19)(16,18)(17,20)(21,22,26,23)(24,27,32,29)(25,30,34,28)(31,36,33,35);(5,6,9,7)(8,10,12,11)(13,15,17,16)(14,18,20,19)(22,23)(27,29)(28,30)(35,36);(2,3)(8,12)(10,11)(13,15,17,16)(14,19,20,18)(21,23,26,22)(24,29,32,27)(25,28,34,30)(31,35,33,36);(1,3,4,2)(5,7,9,6)(8,10,12,11)(13,16,17,15)(14,18,20,19)(21,24,26,32)(22,27,23,29)(25,33,34,31)(28,36,30,35);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,19)(17,20)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fgi,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,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, 4942302);