#4471704

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
automorphism	Aut(G) order 1048576
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

2130 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)(7,11)(8,9)(10,12)(13,14,16,18)(15,19,20,17)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34);(6,9)(10,11)(13,14,16,18)(15,17,20,19)(23,29)(26,33)(30,35)(34,36);(7,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,20)(18,19)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)

Polycyclic presentation:

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

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, 4378900);