#559405

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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

2018 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)(6,7)(8,12)(14,16)(15,17)(21,22)(23,24)(25,29)(26,27)(28,31)(30,32)(33,34)(35,36);(5,6,9,7)(8,11,12,10)(13,14,18,16)(15,20,17,19)(22,27)(24,31)(25,33)(30,35);(1,2,4,3)(5,6,9,7)(8,10,12,11)(13,15,18,17)(14,19,16,20)(21,23,26,28)(22,24,27,31)(25,30,33,35)(29,32,34,36);(1,3,4,2)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,20,17,19)(21,24)(22,28)(23,27)(25,32)(26,31)(29,35)(30,34)(33,36);(5,8)(6,10)(7,11)(9,12)(13,17,18,15)(14,20,16,19)(21,25)(22,29)(23,30)(24,32)(26,33)(27,34)(28,35)(31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,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, 466601);