#191514

order	512
nilpotency class	3
isoclinism	isoclinic class of #19651
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

764 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)(23,30)(24,28)(25,34)(26,29)(27,32)(31,35)(33,36);(5,6,8,10,9,12,15,17)(7,11,14,16,13,18,19,20);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23,25,32,26,33,24,31)(22,27,28,30,29,35,34,36);(7,13)(11,18)(14,19)(16,20)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,30,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98710);