#369041

order	512
nilpotency class	3
isoclinism	isoclinic class of #196299
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276237);