#186786

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×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2070 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)(5,6,11,14,10,16,9,15)(7,12,17,19,13,8,18,20)(21,22)(23,32)(24,33)(25,31)(26,30)(27,29)(28,36)(34,35);(2,4)(5,7)(6,12)(8,16)(9,17)(10,13)(11,18)(14,20)(15,19)(21,23,27,28)(22,24,29,35)(25,34,26,33)(30,36,31,32);(5,8)(6,13)(7,16)(9,19)(10,12)(11,20)(14,18)(15,17)(21,24)(22,28)(23,29)(25,36)(26,32)(27,35)(30,33)(31,34);(5,9,10,11)(6,14,16,15)(7,17,13,18)(8,19,12,20)(22,29)(24,35)(30,31)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,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, 93982);