#189131

order	512
nilpotency class	3
isoclinism	isoclinic class of #19368
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

932 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)(8,15)(11,16)(12,17)(14,19)(18,20)(21,22)(23,31)(24,30)(25,27)(26,29)(28,32)(33,36)(34,35);(5,6,8,10,9,12,15,17)(7,11,14,16,13,18,19,20)(22,27)(28,35)(29,30)(31,36);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23)(22,28)(24,32)(25,33)(26,34)(27,35)(29,31)(30,36);(6,12)(7,13)(10,17)(14,19)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,31,35,36)

Polycyclic presentation:

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

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, 96327);