#366737

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2181 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)(7,11)(8,17)(9,23)(10,15)(12,16)(13,21)(14,24)(18,25)(19,22)(20,28)(26,30)(27,29)(31,33)(32,34)(35,36);(5,7,10,6,12,15,11,16)(8,19,27,13,22,17,21,29)(9,18,26,14,25,23,24,30)(20,32,33,28,35,31,34,36);(2,4)(5,8)(6,13)(7,17)(9,20)(10,21)(11,27)(12,22)(14,28)(15,19)(16,29)(18,31)(23,32)(24,33)(25,35)(26,34)(30,36);(5,9,12,25)(6,14,16,30)(7,18,15,23)(8,20,22,35)(10,26,11,24)(13,28,29,36)(17,31,19,32)(21,34,27,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273933);