#561034

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

2046 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 468230);