#541405

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

1830 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)(5,6,10,9)(7,12,11,8)(13,14,17,16)(15,18,19,20)(21,22,26,23)(24,27,32,29)(25,30,33,28)(31,35,36,34);(5,6,10,9)(7,8,11,12)(15,19)(18,20)(22,23)(25,33)(27,29)(31,36);(5,7)(6,8)(9,12)(10,11)(13,15,17,19)(14,18,16,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,8,10,12)(6,11,9,7)(13,14,17,16)(15,18,19,20)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36);(1,3,4,2)(5,9,10,6)(7,12,11,8)(13,16,17,14)(15,20,19,18)(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=fgh,e^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 448601);