#194173

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1216 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)(5,6,9,14,10,13,8,12)(7,15,16,11,18,19,17,20)(21,22,26,31,27,32,25,30)(23,33,34,28,24,36,35,29);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,19)(13,20)(14,15)(21,23,27,24)(22,28,32,29)(25,34,26,35)(30,36,31,33);(5,8,10,9)(6,12,13,14)(7,16,18,17)(11,19,20,15)(21,24,27,23)(22,29,32,28)(25,35,26,34)(30,33,31,36);(6,13)(11,20)(12,14)(15,19)(21,25,27,26)(22,30,32,31)(23,34,24,35)(28,36,29,33)

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=dhi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101369);