#213169

order	512
nilpotency class	3
isoclinism	isoclinic class of #19713
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

788 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)(5,6)(7,14)(8,12)(9,19)(10,13)(11,17)(15,16)(18,20)(21,22,26,30)(23,31,32,28)(24,27,33,36)(25,34,35,29);(5,7,9,17,10,18,8,16)(6,11,12,14,13,15,19,20)(21,23,25,24,26,32,35,33)(22,27,29,31,30,36,34,28);(5,6)(7,15)(8,19)(9,12)(10,13)(11,18)(14,16)(17,20)(21,24,25,32,26,33,35,23)(22,28,29,27,30,31,34,36);(1,3,4,2)(5,8,10,9)(6,12,13,19)(7,16,18,17)(11,14,15,20)(23,32)(24,33)(27,36)(28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=di,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, 120365);