#4435222

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2051 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)(5,6,9,11)(7,12,10,8)(13,14,17,16)(15,19,20,18)(23,29)(25,33)(27,30)(28,32);(5,7,9,10)(6,8,11,12)(13,14,17,16)(15,18,20,19)(21,22,26,24)(23,30,29,27)(25,28,33,32)(31,36,35,34);(5,6)(7,12)(8,10)(9,11)(13,15,17,20)(14,18,16,19)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36);(1,2,3,4)(5,6)(7,8)(9,11)(10,12)(13,16,17,14)(15,19,20,18)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(5,8)(6,10)(7,11)(9,12)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=fg,d^2=gi,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, 4342418);