#554128

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

1838 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)(5,6,9,10)(7,11,12,8)(13,14)(15,20)(16,17)(18,19)(22,26)(23,29)(28,32)(31,35);(6,10)(8,11)(14,19)(15,17)(21,22,25,26)(23,30,29,27)(24,32,33,28)(31,34,35,36);(1,3,4,2)(13,15,18,17)(14,20,19,16)(21,22)(23,27)(24,28)(25,26)(29,30)(31,34)(32,33)(35,36);(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23,25,29)(22,27,26,30)(24,31,33,35)(28,34,32,36);(5,8)(6,7)(9,11)(10,12)(13,17,18,15)(14,16,19,20)(21,24,25,33)(22,28,26,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

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

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, 461324);