#383763

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3041	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

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

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, 290959);