#580661

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=fi,e^2=h,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, 487857);