#186685

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×#3518	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2238 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)(5,6,10,15,11,14,9,19)(7,8,17,18,13,12,16,20)(22,27)(23,31)(24,25)(29,30)(32,33)(34,35);(2,4)(5,7,11,13)(6,12,14,8)(9,17,10,16)(15,18,19,20)(21,22,26,30)(23,28,32,36)(24,27,25,29)(31,34,33,35);(5,8,11,12)(6,13,14,7)(9,20,10,18)(15,16,19,17)(21,23,26,32)(22,28,30,36)(24,31,25,33)(27,34,29,35);(6,14)(7,13)(15,19)(16,17)(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,c^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, 93881);