#321803

order	512
nilpotency class	3
isoclinism	isoclinic class of #278987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228999);