#222411

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

905 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 129607);