#545717

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2046 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=fh,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 452913);