#386416

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

989 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=efg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi,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, 293612);