#383766

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1350 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ei,d^2=ei,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, 290962);