#369040

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fg,d^2=e,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, 276236);