#577056

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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

1986 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)(7,11)(9,12)(13,14,19,17)(15,20,18,16)(21,22,26,23)(24,29,32,27)(25,28,33,30)(31,35,36,34);(5,6,8,10)(7,9,11,12)(13,15,19,18)(14,20,17,16)(22,23)(27,29)(28,30)(34,35);(5,7)(6,9)(8,11)(10,12)(13,16)(14,18)(15,17)(19,20)(21,23,26,22)(24,29,32,27)(25,28,33,30)(31,34,36,35);(5,6)(7,9)(8,10)(11,12)(13,17,19,14)(15,20,18,16)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,7)(6,9)(8,11)(10,12)(13,18,19,15)(14,20,17,16)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=h,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=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, 484252);