#194243

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1108 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101439);