#4862174

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 2097152
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

2326 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fg,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4769370);