#559806

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fgi,d^2=gi,e^2=gh,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, 467002);