#4308614

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 16777216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2208 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)(7,11)(15,19)(17,20)(22,26)(23,29)(24,31)(34,36);(1,2,4,3)(13,14,16,18)(15,17,19,20)(22,26)(27,32)(28,33)(34,36);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,17,18,20)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215810);