#186769

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
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×#3552	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2204 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)(7,16)(8,17)(9,15)(10,14)(11,13)(12,20)(18,19)(21,22,25,27,26,30,24,29)(23,28,33,34,31,35,32,36);(2,4)(5,7,11,12)(6,8,13,19)(9,18,10,17)(14,20,15,16)(22,27)(24,25)(28,34)(29,30)(32,33)(35,36);(5,8)(6,12)(7,13)(9,20)(10,16)(11,19)(14,17)(15,18)(21,23)(22,28)(24,32)(25,33)(26,31)(27,34)(29,36)(30,35);(6,13)(8,19)(14,15)(16,20)(23,31)(28,35)(32,33)(34,36)

Polycyclic presentation:

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

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, 93965);