#494465

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3060	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1406 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:

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

Polycyclic presentation:

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

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, 401661);