#186742

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2176 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)(9,14)(10,20)(11,15)(12,18)(13,19)(21,22,25,30,26,29,24,28)(23,31,32,27,34,35,33,36);(2,4)(5,7,11,13)(6,12,15,8)(9,18,10,17)(14,19,20,16)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,36)(30,31);(5,8,11,12)(6,13,15,7)(9,19,10,16)(14,18,20,17)(21,24,26,25)(22,28,29,30)(23,32,34,33)(27,35,36,31);(5,9,11,10)(6,14,15,20)(7,17,13,18)(8,19,12,16)(22,29)(27,36)(28,30)(31,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93938);