#247126

order	512
nilpotency class	3
isoclinism	isoclinic class of #247047
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1154 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154322);