#369186

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1830 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)(5,6,11,16,12,17,10,33)(7,18,22,34,23,25,21,13)(8,24,26,14,28,35,27,20)(9,29,31,36,32,19,30,15);(1,2,4,3)(5,7)(6,13,17,34)(8,15,28,36)(9,20)(10,21)(11,22)(12,23)(14,32)(16,18,33,25)(19,26,29,27)(24,31)(30,35);(5,8)(6,14,17,20)(7,19,23,29)(9,25)(10,26)(11,27)(12,28)(13,30)(15,22,36,21)(16,24,33,35)(18,32)(31,34);(1,3,4,2)(5,9,10,30,12,32,11,31)(6,15,16,29,17,36,33,19)(7,20,21,35,23,14,22,24)(8,25,26,13,28,18,27,34)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 276382);