#213133

order	512
nilpotency class	3
isoclinism	isoclinic class of #19651
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

794 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)(23,29)(24,26)(25,33)(27,31)(28,32)(30,35)(34,36);(5,7,9,17,10,18,8,16)(6,11,12,14,13,15,19,20)(22,26)(27,34)(28,33)(29,35);(5,6)(7,15)(8,19)(9,12)(10,13)(11,18)(14,16)(17,20)(21,23,25,31,24,30,32,36)(22,27,28,29,26,34,33,35);(1,3,4,2)(5,8,10,9)(6,12,13,19)(7,16,18,17)(11,14,15,20)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,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^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120329);