#321408

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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 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)(5,6,10,18,8,11,9,13)(7,14,17,12,15,20,16,19)(21,22)(23,26)(24,28)(25,27)(29,33)(30,34)(31,36)(32,35);(1,2,4,3)(6,11)(7,15)(13,18)(16,17)(21,23,29,30,25,32,24,31)(22,26,33,34,27,35,28,36);(5,7,8,15)(6,12,11,19)(9,17,10,16)(13,20,18,14)(23,30)(24,29)(26,34)(28,33)(31,32)(35,36);(1,3,4,2)(6,11)(12,19)(13,18)(14,20)(22,27)(26,35)(28,33)(34,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,c^2=h,d^2=eh,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, 228604);