#383761

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×#535	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1416 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ei,d^2=ei,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,h^b=hi,f^c=fi,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, 290957);