#492867

order	512
nilpotency class	4
isoclinism	isoclinic class of #33404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1586 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)(3,4)(6,12,14,15)(8,21)(9,23)(10,22)(13,24,27,28)(16,30,25,26)(18,32)(19,33)(20,31)(29,36,34,35);(2,4)(5,6,9,15,10,16,23,30,11,14,8,12,22,25,21,26)(7,13,19,28,20,29,33,36,17,27,18,24,31,34,32,35);(5,7)(6,13)(8,18)(9,19)(10,20)(11,17)(12,24)(14,27)(15,28)(16,29)(21,32)(22,31)(23,33)(25,34)(26,35)(30,36);(7,17)(13,27)(18,19)(20,31)(24,28)(29,34)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,f^b=fi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226789, #72226797, #72226798, …
maximal normal subgroup of	#72226789, #72226797, #72226798, …
maximal quotient of	D₄×#1383

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400063);