#494256

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 401452);