#383744

order	512
nilpotency class	3
isoclinism	isoclinic class of #213760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1393 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290940);