#225564

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

665 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)(5,6,10,14)(7,13,9,11)(8,12,16,19)(15,20,17,18)(21,22)(23,27)(24,31)(25,34)(26,29)(28,32)(30,35)(33,36);(6,11)(7,9)(8,15)(12,19)(13,14)(16,17)(21,23,26,25)(22,27,29,34)(24,30,33,32)(28,31,35,36);(5,7,10,9)(6,11,14,13)(8,15,16,17)(12,18,19,20)(21,24,25,32,26,33,23,30)(22,28,27,31,29,35,34,36);(5,8,10,16)(6,12,14,19)(7,15,9,17)(11,18,13,20)

Polycyclic presentation:

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

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, 132760);