#321390

order	512
nilpotency class	3
isoclinism	isoclinic class of #298661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228586);