#353938

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
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	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1926 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)(7,17)(8,22)(9,14)(10,15)(11,16)(12,19)(13,24)(18,32)(20,28)(21,27)(23,29)(25,30)(26,35)(31,33)(34,36);(5,7,10,20,11,21,9,19)(6,12,14,27,16,28,15,17)(8,18,24,34,25,35,23,33)(13,26,22,36,29,32,30,31);(1,3,4,2)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,22)(15,30)(16,29)(17,31)(19,33)(20,34)(21,35)(27,36)(28,32);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,27,28,17)(13,22,29,30)(18,33,35,34)(26,36,32,31)

Polycyclic presentation:

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

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, 261134);