#195630

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
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

1128 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)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20)(21,22,27,29,26,31,25,30)(23,32,34,28,35,36,33,24);(1,2,3,4)(21,23)(22,28)(24,31)(25,34)(26,35)(27,33)(29,32)(30,36);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17)(21,24,26,28)(22,23,31,35)(25,32,27,36)(29,33,30,34);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,25,26,27)(22,29,31,30)(23,33,35,34)(24,32,28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102826);