#369109

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
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

1844 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)(8,18,21,12)(9,22)(10,13)(14,23)(15,24,31,36)(16,26)(17,32)(19,33,35,27)(20,28,29,34)(25,30);(1,2,4,3)(5,7,10,16)(6,11,13,26)(8,19)(9,17,23,30)(12,27)(14,25,22,32)(15,29)(18,33)(20,31)(21,35)(24,34)(28,36);(5,8,9,20,10,21,23,29)(6,12,22,34,13,18,14,28)(7,15,30,35,16,31,17,19)(11,24,25,27,26,36,32,33);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,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, 276305);