#544842

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2096 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=fg,e^2=h,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, 452038);