#186683

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#3515	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2244 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)(5,6,9,14,10,12,20,13)(7,8,16,18,15,11,19,17)(22,27)(23,31)(24,25)(29,30)(32,33)(34,35);(2,4)(5,7,10,15)(6,11,12,8)(9,19,20,16)(13,17,14,18)(21,22,26,30)(23,28,32,36)(24,27,25,29)(31,34,33,35);(5,8)(6,7)(9,18)(10,11)(12,15)(13,19)(14,16)(17,20)(21,23)(22,28)(24,31)(25,33)(26,32)(27,34)(29,35)(30,36);(6,12)(7,15)(13,14)(16,19)(21,24,26,25)(22,29,30,27)(23,31,32,33)(28,35,36,34)

Polycyclic presentation:

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

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, 93879);