#186891

order	512
nilpotency class	3
isoclinism	isoclinic class of #186835
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10)(11,19,22,33)(12,16)(13,18)(14,17)(15,21,29,25)(20,36,38,35)(23,31)(24,30)(26,34)(27,32)(28,37,40,39);(2,5)(3,7)(9,11)(10,15)(12,20)(13,25)(14,22)(16,28)(17,29)(18,33)(19,32)(21,27)(23,39)(24,38)(26,37)(30,40)(31,35)(34,36);(9,12,14,24)(10,16,17,30)(11,20,22,38)(13,26,27,23)(15,28,29,40)(18,34,32,31)(19,35,33,36)(21,39,25,37);(10,17)(15,29)(16,30)(18,32)(19,33)(28,40)(31,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306289, #72306295
maximal normal subgroup of	#72306289, #72306295

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94087);