#213123

order	512
nilpotency class	3
isoclinism	isoclinic class of #19621
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,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=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, 120319);