#545329

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=f,d^2=h,e^2=gh,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, 452525);