#321311

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
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

794 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)(5,6,8,13,9,12,16,11)(7,10,14,19,15,18,20,17)(21,22)(23,26)(24,27)(25,28)(29,36)(30,34)(31,33)(32,35);(1,2,4,3)(5,7,9,15)(6,10,12,18)(8,14,16,20)(11,17,13,19)(21,23,25,31,24,30,32,29)(22,26,28,33,27,34,35,36);(6,11)(8,16)(10,17)(12,13)(14,20)(18,19)(22,27)(23,29)(25,32)(26,33)(30,31)(34,36);(1,3,4,2)(6,12)(10,18)(11,13)(17,19)

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,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, 228507);