#186665

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
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×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2234 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)(5,6,11,14,10,8,9,13)(7,12,18,20,17,15,16,19)(22,26)(23,29)(24,31)(28,32)(30,33)(34,36);(2,4)(5,7,10,17)(6,12,8,15)(9,18,11,16)(13,20,14,19)(21,22,25,28)(23,27,30,35)(24,32,31,26)(29,34,33,36);(5,8,10,6)(7,15,17,12)(9,14,11,13)(16,20,18,19)(21,23,25,30)(22,27,28,35)(24,33,31,29)(26,34,32,36);(5,9,10,11)(6,13,8,14)(7,16,17,18)(12,19,15,20)(22,28)(26,32)(27,35)(34,36)

Polycyclic presentation:

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

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93861);