#222483

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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

898 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)(7,13)(8,16)(11,14)(12,17)(19,20)(21,22)(23,30)(24,28)(25,33)(26,29)(27,32)(31,36)(34,35);(5,6,9,12)(7,14,13,19)(8,17,16,10)(11,18,20,15)(23,31)(27,35)(30,36)(32,34);(21,23)(22,27)(24,32)(25,34)(26,31)(28,36)(29,35)(30,33);(5,7,9,13)(6,11,12,20)(8,18,16,15)(10,19,17,14)(21,24,26,25)(22,28,29,33)(23,32,31,34)(27,36,35,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fi,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, 129679);