#501075

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1067 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408271);