#353976

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1958 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=i,h^2=i,b^a=be,c^a=cf,h^a=hi,e^b=ei,f^b=fi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261172);