#322584

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
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

789 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)(5,6,9,13,8,12,17,11)(7,14,16,19,15,10,18,20);(1,2,4,3)(5,7,8,15)(6,10,12,14)(9,16,17,18)(11,19,13,20)(21,22,26,34,25,29,24,28)(23,30,31,27,33,35,32,36);(6,11)(9,17)(10,19)(12,13)(14,20)(16,18)(21,23,25,33)(22,27,29,36)(24,31,26,32)(28,35,34,30);(1,3,4,2)(6,12)(7,15)(11,13)(16,18)(21,24,25,26)(22,28,29,34)(23,31,33,32)(27,35,36,30)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,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, 229780);