#212916

order	512
nilpotency class	3
isoclinism	isoclinic class of #19733
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#1712	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fhi,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,d^c=dh

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, 120112);